Topological Insulators: Some Basic Concepts

Topological Insulators: Some Basic Concepts

Ion Garate

References for This Lecture

h"p://www.physics.upenn.edu/~kane/

Charlie Kane’s “pedagogical lectures” (highly recommended):

My course notes:

h"p://www.physique.usherbrooke.ca/pages/en/node/8291

…and references therein

Introduction

Insulators (before 1980)

In insulators, an energy gap separates empty and filled electronic states at T=0.

Examples: Covalent insulators (e.g. GaAs), atomic insulators (e.g. solid Ar), vacuum.

Due to the energy gap, insulators do not conduct electricity under weak, time-independent electric fields.

E

k

This talk is about gapped systems that do conduct electricity.

(In contrast, metals have partially filled bands at T=0)

Insulators (after 1980)

m

B

Metallic edge states are robust.

Quantum Hall insulator E

k

Landau levels (LLs)

Hall conductivity is perfectly quantized.

Why is the quantization independent of sample details?

In spite of bulk energy gap, the edges are metallic.

Why can some gapped systems conduct whereas other do not?

Typing Equations for Slides

Ion Garate

May 2, 2014

�xy(B) = n e2

h sgn(B)

�h = �R� · (z ⇥ q) nc ⌘ n" + n# = 0

ns ⌘ n" � n# = 2 sgn(�so)

�sxy ⌘ �"xy � �#xy =

jx" � jx#Ey

= 2

e2

h(1)

n" = sgn(�so)

mH = +�so

mH = ��so

�xy =

e2

h sgn(mH)

mK = mK0

nc = nK + nK0= 0

mK = �mK0

nc = nK + nK0= sgn(m)

|q�i|q+i

⇧(q,!) =

1

V

Xk

Xnn0

|hkn|k� qn0i|2 fkn � fk�qn0

Ekn � Ek�qn0 � !(2)

�Ekn 'Xn0k0

g2q

|hnk|n0k� qi|2Ekn � Ek�qn0

(3)

⇢q =

Xnn0

Xk,k0

h kn|eiq·r| knic†knck0n0(4)

�h = mS�z

�h = mH�z⌧z

h(q) = vx⌧z�xqx + vy�yqy + �so⌧z�zsz

n� =

12⇡

R1 dS · Fq� =

12 sgn(mvxvy)

dS = dqxdqy zdz(k) 6= 0

�xx = �yy = 0

�xy =

e2

h

Xn2occ

1

2⇡

ZBZ

dS · Fkn (5)

1

integer (# of filled LLs)

E

k


m

Quantum Spin Hall insulator

m

Spin-up electrons

Spin-down electrons

Metallic edge states are robust under non-magnetic perturbations

CdTe HgTe CdTe

Made possible by spin-orbit coupling


3D Time-reversal invariant topological insulators

Strong spin-orbit coupling

BiSb, BiTe, BiSe…

Surface states:

Massless Dirac fermions

Energy

Momentum (on surface)

Dirac cones robust under non-magnetic perturbations

kx ky

Energy

Band Theory

Elements of Traditional Band Theory


Ion Garate

April 28, 2014

H| kni = Ekn| kni| kni = eik·r|ukni

= gN0�

TvT · r�⇢y

(1)

2��1e↵ + W 2/D > ⌧�

�e↵ = � exp(�W 2/8l2B)

↵ = 1/2

↵ = �1/2

↵ = 3/2

↵ = 1

2��1+ W 2/D > ⌧�

�⌧� � 1

�⌧� ⌧ 1

⌧ e↵SB = ⌧SB exp(W 2/8l2B)

lB =

p~/eB||

2⌧ e↵SB + W 2/D = ⌧�

�(E) ⇠ ~vF /� ' 65 nm

vF ' 4⇥ 10

5m/s

� ' 4 meV

vkF & �vkF ⌧ �

�zz = 0

�yy = gN0�/T�fk = ��f�k

!(q = vq � i� + �!(q)

!(q) =

r4⇡e2ns

mq + i�q · rT (2)

� = ⇡2��2/T 2

��zz(q, 0) ' 0

vT ⇠ DrT

T cosh

2�

�2T

�(3)

1

Non-interacting electrons moving in a perfectly periodic array of atoms

Hamiltonian

Eigenstate

Eigen-energy

Crystal momentum k is a good quantum number

n is the band label


Ion Garate

April 28, 2014

| kni = eik·r|ukni

= gN0�

TvT · r�⇢y

(1)

2��1e↵ + W 2/D > ⌧�

�e↵ = � exp(�W 2/8l2B)

↵ = 1/2

↵ = �1/2

↵ = 3/2

↵ = 1

2��1+ W 2/D > ⌧�

�⌧� � 1

�⌧� ⌧ 1


lB =

p~/eB||

2⌧ e↵SB + W 2/D = ⌧�

�(E) ⇠ ~vF /� ' 65 nm

vF ' 4⇥ 10

5m/s

� ' 4 meV


�zz = 0

�yy = gN0�/T�fk = ��f�k

!(q = vq � i� + �!(q)

!(q) =

r4⇡e2ns

mq + i�q · rT (2)

� = ⇡2��2/T 2

��zz(q, 0) ' 0

vT ⇠ DrT

T cosh

2�

�2T

�(3)

��yy(q, 0) ' �iN0q · vT

T(4)

1

Bloch’s theorem: Periodic part


Ion Garate

April 28, 2014

h(k) = e�ik·rHeik·r

h(k)|ukni = Ekn|ukniH| kni = Ekn| kni| kni = eik·r|ukni

= gN0�

TvT · r�⇢y

(1)

2��1e↵ + W 2/D > ⌧�

�e↵ = � exp(�W 2/8l2B)

↵ = 1/2

↵ = �1/2

↵ = 3/2

↵ = 1

2��1+ W 2/D > ⌧�

�⌧� � 1

�⌧� ⌧ 1


lB =

p~/eB||

2⌧ e↵SB + W 2/D = ⌧�

�(E) ⇠ ~vF /� ' 65 nm

vF ' 4⇥ 10

5m/s

� ' 4 meV


�zz = 0

�yy = gN0�/T�fk = ��f�k

!(q = vq � i� + �!(q)

!(q) =

r4⇡e2ns

mq + i�q · rT (2)

� = ⇡2��2/T 2

��zz(q, 0) ' 0

1


Ion Garate

April 28, 2014

h(k) = e�ik·rHeik·r


= gN0�

TvT · r�⇢y

(1)

2��1e↵ + W 2/D > ⌧�

�e↵ = � exp(�W 2/8l2B)

↵ = 1/2

↵ = �1/2

↵ = 3/2

↵ = 1

2��1+ W 2/D > ⌧�

�⌧� � 1

�⌧� ⌧ 1


lB =

p~/eB||

2⌧ e↵SB + W 2/D = ⌧�

�(E) ⇠ ~vF /� ' 65 nm

vF ' 4⇥ 10

5m/s

� ' 4 meV


�zz = 0

�yy = gN0�/T�fk = ��f�k

!(q = vq � i� + �!(q)

!(q) =

r4⇡e2ns

mq + i�q · rT (2)

� = ⇡2��2/T 2

��zz(q, 0) ' 0

1

Bloch Hamiltonian :

E

k


Ion Garate

April 29, 2014

ARn = ihuRn|rR|uRni⇡a�⇡

ah(k, t + ⌧) = h(k)

�P =

e

2⇡

ZS

FRn · dS (1)

P =

e

2⇡

Xn2occ

ZBZ

Akn dk (2)

Nn =

1

2⇡

IS

Fkn · dS (3)

FRn = rR ⇥ARn

�n =

HcAkn · dk

�n =

RSFkn · dS

A = �ihukn|rk|ukni|ukni ! ei�k |uknih(k) = e�ik·rHeik·r


= gN0�

TvT · r�⇢y

(4)

2��1e↵ + W 2/D > ⌧�

�e↵ = � exp(�W 2/8l2B)

↵ = 1/2

↵ = �1/2

↵ = 3/2

↵ = 1

2��1+ W 2/D > ⌧�

�⌧� � 1

�⌧� ⌧ 1


lB =

p~/eB||

1


Ion Garate

April 29, 2014


ah(k, t + ⌧) = h(k)

�P =

e

2⇡

ZS

FRn · dS (1)

P =

e

2⇡

Xn2occ

ZBZ

Akn dk (2)

Nn =

1

2⇡

IS

Fkn · dS (3)

FRn = rR ⇥ARn

�n =

HcAkn · dk

�n =

RSFkn · dS



= gN0�

TvT · r�⇢y

(4)

2��1e↵ + W 2/D > ⌧�

�e↵ = � exp(�W 2/8l2B)

↵ = 1/2

↵ = �1/2

↵ = 3/2

↵ = 1

2��1+ W 2/D > ⌧�

�⌧� � 1

�⌧� ⌧ 1


lB =

p~/eB||

1

1st Brillouin zone (BZ)

Energies and wave functions have the periodicity of the reciprocal lattice.

Elements of Topological Band Theory Berry phase:


Ion Garate

May 2, 2014

H(R)|n,Ri = En(R)|n,Ri| (t = 0)i = |n,R(0)i| (t)i = ei✓(t)|n,R(t)i✓(t) =

1~

R t

0dt0En(R(t0))� i

R t

0dt0hnR(t0)| d

dt0 |n,R(t0)iR(t)

�xy(B) = n e2

h sgn(B)

�h = �R� · (z ⇥ q) nc ⌘ n" + n# = 0

ns ⌘ n" � n# = 2 sgn(�so)

�sxy ⌘ �"xy � �#xy =

jx" � jx#Ey

= 2

e2

h(1)

n" = sgn(�so)

mH = +�so

mH = ��so

�xy =

e2

h sgn(mH)

mK = mK0

nc = nK + nK0= 0

mK = �mK0


|q�i|q+i

⇧(q,!) =

1

V

Xk

Xnn0


Ekn � Ek�qn0 � !(2)

�Ekn 'Xn0k0

g2q


(3)

⇢q =

Xnn0

Xk,k0


�h = mS�z

�h = mH�z⌧z


n� =

12⇡

R1 dS · Fq� =

12 sgn(mvxvy)

1

Consider a Hamiltonian that depends on an external (vector) parameter R.


Ion Garate

May 2, 2014


1~

R t

0dt0En(R(t0))� i

R t

0dt0hnR(t0)| d

dt0 |n,R(t0)iR(t)

�xy(B) = n e2

h sgn(B)

�h = �R� · (z ⇥ q) nc ⌘ n" + n# = 0

ns ⌘ n" � n# = 2 sgn(�so)

�sxy ⌘ �"xy � �#xy =

jx" � jx#Ey

= 2

e2

h(1)

n" = sgn(�so)

mH = +�so

mH = ��so

�xy =

e2

h sgn(mH)

mK = mK0

nc = nK + nK0= 0

mK = �mK0


|q�i|q+i

⇧(q,!) =

1

V

Xk

Xnn0


Ekn � Ek�qn0 � !(2)

�Ekn 'Xn0k0

g2q


(3)

⇢q =

Xnn0

Xk,k0


�h = mS�z

�h = mH�z⌧z


n� =

12⇡

R1 dS · Fq� =

12 sgn(mvxvy)

1

Imagine that R changes in time adiabatically.


Ion Garate

May 2, 2014


1~

R t

0dt0En(R(t0))� i

R t

0dt0hnR(t0)| d

dt0 |n,R(t0)iR(t)

�xy(B) = n e2

h sgn(B)

�h = �R� · (z ⇥ q) nc ⌘ n" + n# = 0

ns ⌘ n" � n# = 2 sgn(�so)

�sxy ⌘ �"xy � �#xy =

jx" � jx#Ey

= 2

e2

h(1)

n" = sgn(�so)

mH = +�so

mH = ��so

�xy =

e2

h sgn(mH)

mK = mK0

nc = nK + nK0= 0

mK = �mK0


|q�i|q+i

⇧(q,!) =

1

V

Xk

Xnn0


Ekn � Ek�qn0 � !(2)

�Ekn 'Xn0k0

g2q


(3)

⇢q =

Xnn0

Xk,k0


�h = mS�z

�h = mH�z⌧z


n� =

12⇡

R1 dS · Fq� =

12 sgn(mvxvy)

1


Ion Garate

May 2, 2014

R(t)

�xy(B) = n e2

h sgn(B)

�h = �R� · (z ⇥ q) nc ⌘ n" + n# = 0

ns ⌘ n" � n# = 2 sgn(�so)

�sxy ⌘ �"xy � �#xy =

jx" � jx#Ey

= 2

e2

h(1)

n" = sgn(�so)

mH = +�so

mH = ��so

�xy =

e2

h sgn(mH)

mK = mK0

nc = nK + nK0= 0

mK = �mK0


|q�i|q+i

⇧(q,!) =

1

V

Xk

Xnn0


Ekn � Ek�qn0 � !(2)

�Ekn 'Xn0k0

g2q


(3)

⇢q =

Xnn0

Xk,k0


�h = mS�z

�h = mH�z⌧z


n� =

12⇡

R1 dS · Fq� =

12 sgn(mvxvy)


�xx = �yy = 0

1

C


Ion Garate

May 2, 2014


1~

R t

0dt0En(R(t0))� i

RC

dR · hnR| ddR |n,Ri

R(t)

�xy(B) = n e2

h sgn(B)

�h = �R� · (z ⇥ q) nc ⌘ n" + n# = 0

ns ⌘ n" � n# = 2 sgn(�so)

�sxy ⌘ �"xy � �#xy =

jx" � jx#Ey

= 2

e2

h(1)

n" = sgn(�so)

mH = +�so

mH = ��so

�xy =

e2

h sgn(mH)

mK = mK0

nc = nK + nK0= 0

mK = �mK0


|q�i|q+i

⇧(q,!) =

1

V

Xk

Xnn0


Ekn � Ek�qn0 � !(2)

�Ekn 'Xn0k0

g2q


(3)

⇢q =

Xnn0

Xk,k0


�h = mS�z

�h = mH�z⌧z


n� =

12⇡

R1 dS · Fq� =

12 sgn(mvxvy)

1

Dynamical phase. Berry phase (Geometrical phase)

What is the time evolution of the wave function?

(adiabaticity)

Initial state:

n=eigenstate label.

State at time t:

[M. Berry, 1983]

Elements of Topological Band Theory

Berry connection:

Reminiscent of phase of a particle in an EM field.

The Berry connection is analogue to a vector potential. Thus, it is not gauge-invariant unless C is a closed loop.


Ion Garate

May 3, 2014

HC

An(R) · dR =

RS

dS · [rR ⇥An(R)]

An(R) = hn,R|rR|n,Ri�n =

RC

An(R) · dRH(R)|n,Ri = En(R)|n,Ri| (t = 0)i = |n,R(0)i| (t)i = ei✓(t)|n,R(t)i✓(t) =

1~

R t

0dt0En(R(t0))� i

RC


R(t)

�xy(B) = n e2

h sgn(B)

�h = �R� · (z ⇥ q) nc ⌘ n" + n# = 0

ns ⌘ n" � n# = 2 sgn(�so)

�sxy ⌘ �"xy � �#xy =

jx" � jx#Ey

= 2

e2

h(1)

n" = sgn(�so)

mH = +�so

mH = ��so

�xy =

e2

h sgn(mH)

mK = mK0

nc = nK + nK0= 0

mK = �mK0


|q�i|q+i

⇧(q,!) =

1

V

Xk

Xnn0


Ekn � Ek�qn0 � !(2)

�Ekn 'Xn0k0

g2q


(3)

⇢q =

Xnn0

Xk,k0


�h = mS�z

1


Ion Garate

May 2, 2014

R(t)

�xy(B) = n e2

h sgn(B)

�h = �R� · (z ⇥ q) nc ⌘ n" + n# = 0

ns ⌘ n" � n# = 2 sgn(�so)

�sxy ⌘ �"xy � �#xy =

jx" � jx#Ey

= 2

e2

h(1)

n" = sgn(�so)

mH = +�so

mH = ��so

�xy =

e2

h sgn(mH)

mK = mK0

nc = nK + nK0= 0

mK = �mK0


|q�i|q+i

⇧(q,!) =

1

V

Xk

Xnn0


Ekn � Ek�qn0 � !(2)

�Ekn 'Xn0k0

g2q


(3)

⇢q =

Xnn0

Xk,k0


�h = mS�z

�h = mH�z⌧z


n� =

12⇡

R1 dS · Fq� =

12 sgn(mvxvy)


�xx = �yy = 0

1

C

S Typing Equations for Slides

Ion Garate

May 3, 2014

Fn(R) = rR ⇥An(R)HC

An(R) · dR =

RS

dS · [rR ⇥An(R)]


RC


1~

R t

0dt0En(R(t0))� i

RC


R(t)

�xy(B) = n e2

h sgn(B)

�h = �R� · (z ⇥ q) nc ⌘ n" + n# = 0

ns ⌘ n" � n# = 2 sgn(�so)

�sxy ⌘ �"xy � �#xy =

jx" � jx#Ey

= 2

e2

h(1)

n" = sgn(�so)

mH = +�so

mH = ��so

�xy =

e2

h sgn(mH)

mK = mK0

nc = nK + nK0= 0

mK = �mK0


|q�i|q+i

⇧(q,!) =

1

V

Xk

Xnn0


Ekn � Ek�qn0 � !(2)

�Ekn 'Xn0k0

g2q


(3)

⇢q =

Xnn0

Xk,k0


1


Ion Garate

May 2, 2014


RC


1~

R t

0dt0En(R(t0))� i

RC


R(t)

�xy(B) = n e2

h sgn(B)

�h = �R� · (z ⇥ q) nc ⌘ n" + n# = 0

ns ⌘ n" � n# = 2 sgn(�so)

�sxy ⌘ �"xy � �#xy =

jx" � jx#Ey

= 2

e2

h(1)

n" = sgn(�so)

mH = +�so

mH = ��so

�xy =

e2

h sgn(mH)

mK = mK0

nc = nK + nK0= 0

mK = �mK0


|q�i|q+i

⇧(q,!) =

1

V

Xk

Xnn0


Ekn � Ek�qn0 � !(2)

�Ekn 'Xn0k0

g2q


(3)

⇢q =

Xnn0

Xk,k0


�h = mS�z

�h = mH�z⌧z

1

Stokes’ theorem:

Berry curvature:

Berry phase:

The Berry curvature is analogue to a magnetic field. Thus, it is gauge-invariant


Ion Garate

May 6, 2014

An(R) = ihn,R|rR|n,Ri� = 0

� = 1

d|Eg|/dT < 0

d|Eg|/dT > 0

0nkn0

�so < 1mK

nK = nK0

nK = �nK0

nChern = nK + nK0= sgn(mH)

|+i|�inK =

12 sgn(vx

KvyKmK)

dz(q) = mS dz(q) = mH⌧z

|dz(0)|h(q) = v ⌧z�x qx + v �yqy + dz(q)�z

h(d) = d · �|±iE± = ±|d|!c = eB/(mc)c py/(eB)

�⇡/a ⇡/a h(k) = dx(k)�x+ dy(k)�y

Fn(R) = rR ⇥An(R)

�P =

e

2⇡

Xn2occ

IC

An(R) · dR =

e

2⇡

Xn2occ

ZS

Fn(R) · dS = nChern ⇥ e (1)

An(R) = ihun(R)|rR|un(R)ih(k, t) = h(k, t + ⌧)

R = (k, t)

Q =

Z ⌧

0

dt@P

@t= P (t = ⌧)� P (t = 0) ⌘ �P (2)

j = @P/@t

1


Analogies:

The Chern number is an integer. It is also a topological invariant, i.e. independent of the details of the Hamiltonian.

Chern number:

S S Consider a closed surface S, such as


Ion Garate

May 3, 2014

nChern =

12⇡

RSFn(R) · dS


An(R) · dR =

RS

dS · [rR ⇥An(R)]


RC


1~

R t

0dt0En(R(t0))� i

RC


R(t)

�xy(B) = n e2

h sgn(B)

�h = �R� · (z ⇥ q) nc ⌘ n" + n# = 0

ns ⌘ n" � n# = 2 sgn(�so)

�sxy ⌘ �"xy � �#xy =

jx" � jx#Ey

= 2

e2

h(1)

n" = sgn(�so)

mH = +�so

mH = ��so

�xy =

e2

h sgn(mH)

mK = mK0

nc = nK + nK0= 0

mK = �mK0


|q�i|q+i

⇧(q,!) =

1

V

Xk

Xnn0


Ekn � Ek�qn0 � !(2)

�Ekn 'Xn0k0

g2q


(3)

⇢q =

Xnn0

Xk,k0


1

or Typing Equations for Slides

Ion Garate

May 3, 2014

HS✏E · dS = e⇥ integer

nChern =

12⇡

RSFn(R) · dS


An(R) · dR =

RS

dS · [rR ⇥An(R)]


RC


1~

R t

0dt0En(R(t0))� i

RC


R(t)

�xy(B) = n e2

h sgn(B)

�h = �R� · (z ⇥ q) nc ⌘ n" + n# = 0

ns ⌘ n" � n# = 2 sgn(�so)

�sxy ⌘ �"xy � �#xy =

jx" � jx#Ey

= 2

e2

h(1)

n" = sgn(�so)

mH = +�so

mH = ��so

�xy =

e2

h sgn(mH)

mK = mK0

nc = nK + nK0= 0

mK = �mK0


|q�i|q+i

⇧(q,!) =

1

V

Xk

Xnn0


Ekn � Ek�qn0 � !(2)

�Ekn 'Xn0k0

g2q


(3)

1

Gauss’ Law:

Gauss-Bonnet theorem:


Ion Garate

May 3, 2014

14⇡

HS dS = 1� gH

S✏E · dS = e⇥ integer

nChern =

12⇡

RSFn(R) · dS


An(R) · dR =

RS

dS · [rR ⇥An(R)]


RC


1~

R t

0dt0En(R(t0))� i

RC


R(t)

�xy(B) = n e2

h sgn(B)

�h = �R� · (z ⇥ q) nc ⌘ n" + n# = 0

ns ⌘ n" � n# = 2 sgn(�so)

�sxy ⌘ �"xy � �#xy =

jx" � jx#Ey

= 2

e2

h(1)

n" = sgn(�so)

mH = +�so

mH = ��so

�xy =

e2

h sgn(mH)

mK = mK0

nc = nK + nK0= 0

mK = �mK0


|q�i|q+i

⇧(q,!) =

1

V

Xk

Xnn0


Ekn � Ek�qn0 � !(2)

�Ekn 'Xn0k0

g2q


(3)

1

“Genus”= number of holes Gaussian curvature

Electric field Electron’s charge


Example: two-level system (e.g. spin ½ particle in a magnetic field)

Berry PhasePhase ambiguity of quantum mechanical wave function

( )( ) ( )iu e u kk kBerry connection : like a vector potential ( ) ( )i u u kA k k

( ) kA A k

Berry phase : change in phase on a closed loop C C Cd A k

Berry curvature : kF A 2C S

d k FFamous example : eigenstates of 2 level Hamiltonian

( ) ( ) z x y

x y z

d d idH

d id d

k d k

( ) ( ) ( ) ( )H u u k k d k k 1 ˆSolid Angle swept out by ( )2C d k

Cd

S

Field of a monopole located at band degeneracy points.

Eigenstates: Eigenvalues:

Berry phase = ½ ( solid angle subtended by C)

Chern number=monopole charge inside closed surface

Vector d plays the role of parameter R


Ion Garate

May 4, 2014


�⇡/a ⇡/a h(k) = dx(k)�x+ dy(k)�y

Fn(R) = rR ⇥An(R)

�P =

e

2⇡

Xn2occ

IC

An(R) · dR =

e

2⇡

Xn2occ

ZS


An(R) = ihun(R)|rR|un(R)ih(k, t) = h(k, t + ⌧)R = (k, t)

Q =

Z ⌧

0

dt@P

@t= P (t = ⌧)� P (t = 0) ⌘ �P (2)

j = @P/@t⇢ = �@xP� = P · n14⇡

HS dS = 1� gH


nChern =

12⇡

RSFn(R) · dS


An(R) · dR =

RS

dS · [rR ⇥An(R)]


RC


1~

R t

0dt0En(R(t0))� i

RC


R(t)

�xy(B) = n e2

h sgn(B)

�h = �R� · (z ⇥ q) nc ⌘ n" + n# = 0

ns ⌘ n" � n# = 2 sgn(�so)

1


Ion Garate

May 4, 2014


�⇡/a ⇡/a h(k) = dx(k)�x+ dy(k)�y

Fn(R) = rR ⇥An(R)

�P =

e

2⇡

Xn2occ

IC

An(R) · dR =

e

2⇡

Xn2occ

ZS



Q =

Z ⌧

0

dt@P

@t= P (t = ⌧)� P (t = 0) ⌘ �P (2)

j = @P/@t⇢ = �@xP� = P · n14⇡

HS dS = 1� gH


nChern =

12⇡

RSFn(R) · dS


An(R) · dR =

RS

dS · [rR ⇥An(R)]


RC


1~

R t

0dt0En(R(t0))� i

RC


R(t)

�xy(B) = n e2

h sgn(B)

�h = �R� · (z ⇥ q) nc ⌘ n" + n# = 0

ns ⌘ n" � n# = 2 sgn(�so)

1


Ion Garate

May 4, 2014


�⇡/a ⇡/a h(k) = dx(k)�x+ dy(k)�y

Fn(R) = rR ⇥An(R)

�P =

e

2⇡

Xn2occ

IC

An(R) · dR =

e

2⇡

Xn2occ

ZS



Q =

Z ⌧

0

dt@P

@t= P (t = ⌧)� P (t = 0) ⌘ �P (2)

j = @P/@t⇢ = �@xP� = P · n14⇡

HS dS = 1� gH


nChern =

12⇡

RSFn(R) · dS


An(R) · dR =

RS

dS · [rR ⇥An(R)]


RC


1~

R t

0dt0En(R(t0))� i

RC


R(t)

�xy(B) = n e2

h sgn(B)

�h = �R� · (z ⇥ q) nc ⌘ n" + n# = 0

ns ⌘ n" � n# = 2 sgn(�so)

1


Ion Garate

April 30, 2014

ARn = ihuRn|rR|uRnidx(k) = (t + �t) + (t� �t) cos(ka)

dy(k) = (t� �t) sin(ka)

dz(k) = 0

� = 0

� = ⇡�t > 0

�t < 0

|k±iEk± = ±|d(k)|F± = ± d

2d3

h(k) = d(k) · �

�� = �i

IC

hk� | @

@k|k�i · dk = �i

IC0h�| @

@d|�i · dd (1)

h(d) = d · �h(k, t + ⌧) = h(k)

�P =

e

2⇡

Xn2occ

ZS

FRn · dS (2)

P =

e

2⇡

Xn2occ

ZBZ

Akn dk (3)

Nn =

1

2⇡

IS

Fkn · dS (4)

FRn = rR ⇥ARn

�n =

HcAkn · dk

�n =

RSFkn · dS



1

Berry curvature:

Band Topology in 1D

Electric Polarization of a 1D Insulator

Electrical polarization defined only modulo

+ - + + - - + -

Electric polarization = dipole moment per unit volume.


Ion Garate

May 3, 2014

⇢ = �@xP14⇡

HS dS = 1� gH


nChern =

12⇡

RSFn(R) · dS


An(R) · dR =

RS

dS · [rR ⇥An(R)]


RC


1~

R t

0dt0En(R(t0))� i

RC


R(t)

�xy(B) = n e2

h sgn(B)

�h = �R� · (z ⇥ q) nc ⌘ n" + n# = 0

ns ⌘ n" � n# = 2 sgn(�so)

�sxy ⌘ �"xy � �#xy =

jx" � jx#Ey

= 2

e2

h(1)

n" = sgn(�so)

mH = +�so

mH = ��so

�xy =

e2

h sgn(mH)

mK = mK0

nc = nK + nK0= 0

mK = �mK0


|q�i|q+i

⇧(q,!) =

1

V

Xk

Xnn0


Ekn � Ek�qn0 � !(2)

�Ekn 'Xn0k0

g2q


(3)

1

Polarization charge


Ion Garate

May 3, 2014

⇢ = �@xP� = P · n14⇡

HS dS = 1� gH


nChern =

12⇡

RSFn(R) · dS


An(R) · dR =

RS

dS · [rR ⇥An(R)]


RC


1~

R t

0dt0En(R(t0))� i

RC


R(t)

�xy(B) = n e2

h sgn(B)

�h = �R� · (z ⇥ q) nc ⌘ n" + n# = 0

ns ⌘ n" � n# = 2 sgn(�so)

�sxy ⌘ �"xy � �#xy =

jx" � jx#Ey

= 2

e2

h(1)

n" = sgn(�so)

mH = +�so

mH = ��so

�xy =

e2

h sgn(mH)

mK = mK0

nc = nK + nK0= 0

mK = �mK0


|q�i|q+i

⇧(q,!) =

1

V

Xk

Xnn0


Ekn � Ek�qn0 � !(2)

1

Polarization current


Ion Garate

May 3, 2014

j = @P/@t⇢ = �@xP� = P · n14⇡

HS dS = 1� gH


nChern =

12⇡

RSFn(R) · dS


An(R) · dR =

RS

dS · [rR ⇥An(R)]


RC


1~

R t

0dt0En(R(t0))� i

RC


R(t)

�xy(B) = n e2

h sgn(B)

�h = �R� · (z ⇥ q) nc ⌘ n" + n# = 0

ns ⌘ n" � n# = 2 sgn(�so)

�sxy ⌘ �"xy � �#xy =

jx" � jx#Ey

= 2

e2

h(1)

n" = sgn(�so)

mH = +�so

mH = ��so

�xy =

e2

h sgn(mH)

mK = mK0

nc = nK + nK0= 0

mK = �mK0


|q�i|q+i

⇧(q,!) =

1

V

Xk

Xnn0


Ekn � Ek�qn0 � !(2)

1

……


Ion Garate

May 5, 2014

�so < 1mK

nK = nK0

nK = �nK0


|+i|�inK =

12 sgn(vx

KvyKmK)




�⇡/a ⇡/a h(k) = dx(k)�x+ dy(k)�y

Fn(R) = rR ⇥An(R)

�P =

e

2⇡

Xn2occ

IC

An(R) · dR =

e

2⇡

Xn2occ

ZS



R = (k, t)

Q =

Z ⌧

0

dt@P

@t= P (t = ⌧)� P (t = 0) ⌘ �P (2)

j = @P/@t⇢ = �@xP� = P · n14⇡

HS

dS = 1� gHS

✏E · dS = e⇥ integer

nChern =

12⇡

RSFn(R) · dS


An(R) · dR =

RS

dS · [rR ⇥An(R)]

1


Ion Garate

May 5, 2014

�so < 1mK

nK = nK0

nK = �nK0


|+i|�inK =

12 sgn(vx

KvyKmK)




�⇡/a ⇡/a h(k) = dx(k)�x+ dy(k)�y

Fn(R) = rR ⇥An(R)

�P =

e

2⇡

Xn2occ

IC

An(R) · dR =

e

2⇡

Xn2occ

ZS



R = (k, t)

Q =

Z ⌧

0

dt@P

@t= P (t = ⌧)� P (t = 0) ⌘ �P (2)

j = @P/@t⇢ = �@xP� = P · n14⇡

HS

dS = 1� gHS


nChern =

12⇡

RSFn(R) · dS


An(R) · dR =

RS

dS · [rR ⇥An(R)]

1

E

k


Ion Garate

April 29, 2014


ah(k, t + ⌧) = h(k)

�P =

e

2⇡

ZS

FRn · dS (1)

P =

e

2⇡

Xn2occ

ZBZ

Akn dk (2)

Nn =

1

2⇡

IS

Fkn · dS (3)

FRn = rR ⇥ARn

�n =

HcAkn · dk

�n =

RSFkn · dS



= gN0�

TvT · r�⇢y

(4)

2��1e↵ + W 2/D > ⌧�

�e↵ = � exp(�W 2/8l2B)

↵ = 1/2

↵ = �1/2

↵ = 3/2

↵ = 1

2��1+ W 2/D > ⌧�

�⌧� � 1

�⌧� ⌧ 1


lB =

p~/eB||

1


Ion Garate

April 29, 2014


ah(k, t + ⌧) = h(k)

�P =

e

2⇡

ZS

FRn · dS (1)

P =

e

2⇡

Xn2occ

ZBZ

Akn dk (2)

Nn =

1

2⇡

IS

Fkn · dS (3)

FRn = rR ⇥ARn

�n =

HcAkn · dk

�n =

RSFkn · dS



= gN0�

TvT · r�⇢y

(4)

2��1e↵ + W 2/D > ⌧�

�e↵ = � exp(�W 2/8l2B)

↵ = 1/2

↵ = �1/2

↵ = 3/2

↵ = 1

2��1+ W 2/D > ⌧�

�⌧� � 1

�⌧� ⌧ 1


lB =

p~/eB||

1

h(k, t + ⌧) = h(k)

�P =

e

2⇡

Xn2occ

ZS

FRn · dS (8)

P =

e

2⇡

Xn2occ

ZBZ

An(k) dk (9)

An(k) = ihukn| @@k |ukni

Nn =

1

2⇡

IS

Fkn · dS (10)

FRn = rR ⇥ARn

�n =

HcAkn · dk

�n =

RSFkn · dS



= gN0�

TvT · r�⇢y

(11)

2��1e↵ + W 2/D > ⌧�

�e↵ = � exp(�W 2/8l2B)

↵ = 1/2

↵ = �1/2

↵ = 3/2

↵ = 1

2��1+ W 2/D > ⌧�

�⌧� � 1

�⌧� ⌧ 1


lB =

p~/eB||

2⌧ e↵SB + W 2/D = ⌧�

�(E) ⇠ ~vF /� ' 65 nm

vF ' 4⇥ 10

5m/s

� ' 4 meV


�zz = 0

�yy = gN0�/T�fk = ��f�k

!(q = vq � i� + �!(q)

3

h(k, t + ⌧) = h(k)

�P =

e

2⇡

Xn2occ

ZS

FRn · dS (8)

P =

e

2⇡

Xn2occ

ZBZ

An(k) dk (9)

An(k) = ihukn| @@k |ukni

Nn =

1

2⇡

IS

Fkn · dS (10)

FRn = rR ⇥ARn

�n =

HcAkn · dk

�n =

RSFkn · dS



= gN0�

TvT · r�⇢y

(11)

2��1e↵ + W 2/D > ⌧�

�e↵ = � exp(�W 2/8l2B)

↵ = 1/2

↵ = �1/2

↵ = 3/2

↵ = 1

2��1+ W 2/D > ⌧�

�⌧� � 1

�⌧� ⌧ 1


lB =

p~/eB||

2⌧ e↵SB + W 2/D = ⌧�

�(E) ⇠ ~vF /� ' 65 nm

vF ' 4⇥ 10

5m/s

� ' 4 meV


�zz = 0

�yy = gN0�/T�fk = ��f�k

!(q = vq � i� + �!(q)

3

Link between quantum mechanics and classical electrostatics.

Relation between electric polarization and Berry phase Typing Equations for Slides

Ion Garate

April 29, 2014


ah(k, t + ⌧) = h(k)

�P =

e

2⇡

ZS

FRn · dS (1)

P =

e

2⇡

Xn2occ

ZBZ

Akn dk (2)

Nn =

1

2⇡

IS

Fkn · dS (3)

FRn = rR ⇥ARn

�n =

HcAkn · dk

�n =

RSFkn · dS



= gN0�

TvT · r�⇢y

(4)

2��1e↵ + W 2/D > ⌧�

�e↵ = � exp(�W 2/8l2B)

↵ = 1/2

↵ = �1/2

↵ = 3/2

↵ = 1

2��1+ W 2/D > ⌧�

�⌧� � 1

�⌧� ⌧ 1


lB =

p~/eB||

1


Ion Garate

April 29, 2014


ah(k, t + ⌧) = h(k)

�P =

e

2⇡

ZS

FRn · dS (1)

P =

e

2⇡

Xn2occ

ZBZ

Akn dk (2)

Nn =

1

2⇡

IS

Fkn · dS (3)

FRn = rR ⇥ARn

�n =

HcAkn · dk

�n =

RSFkn · dS



= gN0�

TvT · r�⇢y

(4)

2��1e↵ + W 2/D > ⌧�

�e↵ = � exp(�W 2/8l2B)

↵ = 1/2

↵ = �1/2

↵ = 3/2

↵ = 1

2��1+ W 2/D > ⌧�

�⌧� � 1

�⌧� ⌧ 1


lB =

p~/eB||

1

k

k


Ion Garate

April 29, 2014


ah(k, t + ⌧) = h(k)

�P =

e

2⇡

ZS

FRn · dS (1)

P =

e

2⇡

Xn2occ

ZBZ

Akn dk (2)

Nn =

1

2⇡

IS

Fkn · dS (3)

FRn = rR ⇥ARn

�n =

HcAkn · dk

�n =

RSFkn · dS



= gN0�

TvT · r�⇢y

(4)

2��1e↵ + W 2/D > ⌧�

�e↵ = � exp(�W 2/8l2B)

↵ = 1/2

↵ = �1/2

↵ = 3/2

↵ = 1

2��1+ W 2/D > ⌧�

�⌧� � 1

�⌧� ⌧ 1


lB =

p~/eB||

1


Ion Garate

April 29, 2014


ah(k, t + ⌧) = h(k)

�P =

e

2⇡

ZS

FRn · dS (1)

P =

e

2⇡

Xn2occ

ZBZ

Akn dk (2)

Nn =

1

2⇡

IS

Fkn · dS (3)

FRn = rR ⇥ARn

�n =

HcAkn · dk

�n =

RSFkn · dS



= gN0�

TvT · r�⇢y

(4)

2��1e↵ + W 2/D > ⌧�

�e↵ = � exp(�W 2/8l2B)

↵ = 1/2

↵ = �1/2

↵ = 3/2

↵ = 1

2��1+ W 2/D > ⌧�

�⌧� � 1

�⌧� ⌧ 1


lB =

p~/eB||

1

=

[Resta & Vanderbilt, 1994]

Berry connection. k plays the role of R

Berry phase defined modulo à


Ion Garate

May 3, 2014

j = @P/@t⇢ = �@xP� = P · n14⇡

HS dS = 1� gH


nChern =

12⇡

RSFn(R) · dS


An(R) · dR =

RS

dS · [rR ⇥An(R)]


RC


1~

R t

0dt0En(R(t0))� i

RC


R(t)

�xy(B) = n e2

h sgn(B)

�h = �R� · (z ⇥ q) nc ⌘ n" + n# = 0

ns ⌘ n" � n# = 2 sgn(�so)

�sxy ⌘ �"xy � �#xy =

jx" � jx#Ey

= 2

e2

h(1)

n" = sgn(�so)

mH = +�so

mH = ��so

�xy =

e2

h sgn(mH)

mK = mK0

nc = nK + nK0= 0

mK = �mK0


|q�i|q+i

⇧(q,!) =

1

V

Xk

Xnn0


Ekn � Ek�qn0 � !(2)

1

defined modulo


Ion Garate

May 3, 2014

⇢ = �@xP� = P · n14⇡

HS dS = 1� gH


nChern =

12⇡

RSFn(R) · dS


An(R) · dR =

RS

dS · [rR ⇥An(R)]


RC


1~

R t

0dt0En(R(t0))� i

RC


R(t)

�xy(B) = n e2

h sgn(B)

�h = �R� · (z ⇥ q) nc ⌘ n" + n# = 0

ns ⌘ n" � n# = 2 sgn(�so)

�sxy ⌘ �"xy � �#xy =

jx" � jx#Ey

= 2

e2

h(1)

n" = sgn(�so)

mH = +�so

mH = ��so

�xy =

e2

h sgn(mH)

mK = mK0

nc = nK + nK0= 0

mK = �mK0


|q�i|q+i

⇧(q,!) =

1

V

Xk

Xnn0


Ekn � Ek�qn0 � !(2)

1


Ion Garate

May 6, 2014

An(R) = ihn,R|rR|n,Ri� = 0

� = 1

d|Eg|/dT < 0

d|Eg|/dT > 0

0nkn0

�so < 1mK

nK = nK0

nK = �nK0


|+i|�inK =

12 sgn(vx

KvyKmK)




�⇡/a ⇡/a h(k) = dx(k)�x+ dy(k)�y

Fn(R) = rR ⇥An(R)

�P =

e

2⇡

Xn2occ

IC

An(R) · dR =

e

2⇡

Xn2occ

ZS



R = (k, t)

Q =

Z ⌧

0

dt@P

@t= P (t = ⌧)� P (t = 0) ⌘ �P (2)

j = @P/@t

1


Ion Garate

May 4, 2014

Fn(R) = rR ⇥An(R)

�P =

e

2⇡

Xn2occ

IC

An(R) · dR =

e

2⇡

Xn2occ

ZS



Q =

Z ⌧

0

dt@P

@t= P (t = ⌧)� P (t = 0) ⌘ �P (2)

j = @P/@t⇢ = �@xP� = P · n14⇡

HS dS = 1� gH


nChern =

12⇡

RSFn(R) · dS


An(R) · dR =

RS

dS · [rR ⇥An(R)]


RC


1~

R t

0dt0En(R(t0))� i

RC


R(t)

�xy(B) = n e2

h sgn(B)

�h = �R� · (z ⇥ q) nc ⌘ n" + n# = 0

ns ⌘ n" � n# = 2 sgn(�so)

�sxy ⌘ �"xy � �#xy =

jx" � jx#Ey

= 2

e2

h(3)

n" = sgn(�so)

mH = +�so

1

Thouless Charge Pump (1983)

What is the charge pumped in one cycle of the external perturbation?

t

k


Ion Garate

April 29, 2014


ah(k, t + ⌧) = h(k)

�P =

e

2⇡

ZS

FRn · dS (1)

P =

e

2⇡

Xn2occ

ZBZ

Akn dk (2)

Nn =

1

2⇡

IS

Fkn · dS (3)

FRn = rR ⇥ARn

�n =

HcAkn · dk

�n =

RSFkn · dS



= gN0�

TvT · r�⇢y

(4)

2��1e↵ + W 2/D > ⌧�

�e↵ = � exp(�W 2/8l2B)

↵ = 1/2

↵ = �1/2

↵ = 3/2

↵ = 1

2��1+ W 2/D > ⌧�

�⌧� � 1

�⌧� ⌧ 1


lB =

p~/eB||

1


Ion Garate

April 29, 2014


ah(k, t + ⌧) = h(k)

�P =

e

2⇡

ZS

FRn · dS (1)

P =

e

2⇡

Xn2occ

ZBZ

Akn dk (2)

Nn =

1

2⇡

IS

Fkn · dS (3)

FRn = rR ⇥ARn

�n =

HcAkn · dk

�n =

RSFkn · dS



= gN0�

TvT · r�⇢y

(4)

2��1e↵ + W 2/D > ⌧�

�e↵ = � exp(�W 2/8l2B)

↵ = 1/2

↵ = �1/2

↵ = 3/2

↵ = 1

2��1+ W 2/D > ⌧�

�⌧� � 1

�⌧� ⌧ 1


lB =

p~/eB||

1


Ion Garate

April 29, 2014

ARn = ihuRn|rR|uRni⇡a⌧h(k, t + ⌧) = h(k)

�P =

e

2⇡

ZS

FRn · dS (1)

P =

e

2⇡

Xn2occ

ZBZ

Akn dk (2)

Nn =

1

2⇡

IS

Fkn · dS (3)

FRn = rR ⇥ARn

�n =

HcAkn · dk

�n =

RSFkn · dS



= gN0�

TvT · r�⇢y

(4)

2��1e↵ + W 2/D > ⌧�

�e↵ = � exp(�W 2/8l2B)

↵ = 1/2

↵ = �1/2

↵ = 3/2

↵ = 1

2��1+ W 2/D > ⌧�

�⌧� � 1

�⌧� ⌧ 1


lB =

p~/eB||

1

1D insulator under a time-periodic perturbation.


Ion Garate

May 3, 2014

Q =

Z ⌧

0

dt@P

@t= P (t = ⌧)� P (t = 0) ⌘ �P (1)

j = @P/@t⇢ = �@xP� = P · n14⇡

HS dS = 1� gH


nChern =

12⇡

RSFn(R) · dS


An(R) · dR =

RS

dS · [rR ⇥An(R)]


RC


1~

R t

0dt0En(R(t0))� i

RC


R(t)

�xy(B) = n e2

h sgn(B)

�h = �R� · (z ⇥ q) nc ⌘ n" + n# = 0

ns ⌘ n" � n# = 2 sgn(�so)

�sxy ⌘ �"

xy � �#xy =

jx" � jx#Ey

= 2

e2

h(2)

n" = sgn(�so)

mH = +�so

mH = ��so

�xy =

e2

h sgn(mH)

mK = mK0

nc = nK + nK0= 0

mK = �mK0


|q�i|q+i

1

Bloch Hamiltonian:


Ion Garate

May 4, 2014

h(k, t) = h(k, t + ⌧)

Q =

Z ⌧

0

dt@P

@t= P (t = ⌧)� P (t = 0) ⌘ �P (1)

j = @P/@t⇢ = �@xP� = P · n14⇡

HS dS = 1� gH


nChern =

12⇡

RSFn(R) · dS


An(R) · dR =

RS

dS · [rR ⇥An(R)]


RC


1~

R t

0dt0En(R(t0))� i

RC


R(t)

�xy(B) = n e2

h sgn(B)

�h = �R� · (z ⇥ q) nc ⌘ n" + n# = 0

ns ⌘ n" � n# = 2 sgn(�so)

�sxy ⌘ �"

xy � �#xy =

jx" � jx#Ey

= 2

e2

h(2)

n" = sgn(�so)

mH = +�so

mH = ��so

�xy =

e2

h sgn(mH)

mK = mK0

nc = nK + nK0= 0

mK = �mK0


|q�i

1


Ion Garate

May 4, 2014

h(k, t) = h(k, t + ⌧)R = (k, t)

Q =

Z ⌧

0

dt@P

@t= P (t = ⌧)� P (t = 0) ⌘ �P (1)

j = @P/@t⇢ = �@xP� = P · n14⇡

HS dS = 1� gH


nChern =

12⇡

RSFn(R) · dS


An(R) · dR =

RS

dS · [rR ⇥An(R)]


RC


1~

R t

0dt0En(R(t0))� i

RC


R(t)

�xy(B) = n e2

h sgn(B)

�h = �R� · (z ⇥ q) nc ⌘ n" + n# = 0

ns ⌘ n" � n# = 2 sgn(�so)

�sxy ⌘ �"

xy � �#xy =

jx" � jx#Ey

= 2

e2

h(2)

n" = sgn(�so)

mH = +�so

mH = ��so

�xy =

e2

h sgn(mH)

mK = mK0

nc = nK + nK0= 0

mK = �mK0


1

S S


Ion Garate

April 29, 2014


�P =

e

2⇡

ZS

FRn · dS (1)

P =

e

2⇡

Xn2occ

ZBZ

Akn dk (2)

Nn =

1

2⇡

IS

Fkn · dS (3)

FRn = rR ⇥ARn

�n =

HcAkn · dk

�n =

RSFkn · dS



= gN0�

TvT · r�⇢y

(4)

2��1e↵ + W 2/D > ⌧�

�e↵ = � exp(�W 2/8l2B)

↵ = 1/2

↵ = �1/2

↵ = 3/2

↵ = 1

2��1+ W 2/D > ⌧�

�⌧� � 1

�⌧� ⌧ 1


lB =

p~/eB||

1

0

C

Su-Schrieffer-Heeger Model (1979)


Ion Garate

April 29, 2014

ARn = ihuRn|rR|uRni⇡at + �tt� �th(k, t + ⌧) = h(k)

�P =

e

2⇡

Xn2occ

ZS

FRn · dS (1)

P =

e

2⇡

Xn2occ

ZBZ

Akn dk (2)

Nn =

1

2⇡

IS

Fkn · dS (3)

FRn = rR ⇥ARn

�n =

HcAkn · dk

�n =

RSFkn · dS



= gN0�

TvT · r�⇢y

(4)

2��1e↵ + W 2/D > ⌧�

�e↵ = � exp(�W 2/8l2B)

↵ = 1/2

↵ = �1/2

↵ = 3/2

↵ = 1

2��1+ W 2/D > ⌧�

�⌧� � 1

�⌧� ⌧ 1


1


Ion Garate

April 29, 2014

ARn = ihuRn|rR|uRni⇡at + �tt� �th(k, t + ⌧) = h(k)

�P =

e

2⇡

Xn2occ

ZS

FRn · dS (1)

P =

e

2⇡

Xn2occ

ZBZ

Akn dk (2)

Nn =

1

2⇡

IS

Fkn · dS (3)

FRn = rR ⇥ARn

�n =

HcAkn · dk

�n =

RSFkn · dS



= gN0�

TvT · r�⇢y

(4)

2��1e↵ + W 2/D > ⌧�

�e↵ = � exp(�W 2/8l2B)

↵ = 1/2

↵ = �1/2

↵ = 3/2

↵ = 1

2��1+ W 2/D > ⌧�

�⌧� � 1

�⌧� ⌧ 1


1

1D lattice, spinless electrons, half filling. E

k


Ion Garate

April 29, 2014


ah(k, t + ⌧) = h(k)

�P =

e

2⇡

ZS

FRn · dS (1)

P =

e

2⇡

Xn2occ

ZBZ

Akn dk (2)

Nn =

1

2⇡

IS

Fkn · dS (3)

FRn = rR ⇥ARn

�n =

HcAkn · dk

�n =

RSFkn · dS



= gN0�

TvT · r�⇢y

(4)

2��1e↵ + W 2/D > ⌧�

�e↵ = � exp(�W 2/8l2B)

↵ = 1/2

↵ = �1/2

↵ = 3/2

↵ = 1

2��1+ W 2/D > ⌧�

�⌧� � 1

�⌧� ⌧ 1


lB =

p~/eB||

1


Ion Garate

April 29, 2014


ah(k, t + ⌧) = h(k)

�P =

e

2⇡

ZS

FRn · dS (1)

P =

e

2⇡

Xn2occ

ZBZ

Akn dk (2)

Nn =

1

2⇡

IS

Fkn · dS (3)

FRn = rR ⇥ARn

�n =

HcAkn · dk

�n =

RSFkn · dS



= gN0�

TvT · r�⇢y

(4)

2��1e↵ + W 2/D > ⌧�

�e↵ = � exp(�W 2/8l2B)

↵ = 1/2

↵ = �1/2

↵ = 3/2

↵ = 1

2��1+ W 2/D > ⌧�

�⌧� � 1

�⌧� ⌧ 1


lB =

p~/eB||

1


Ion Garate

April 30, 2014



dz(k) = 0

� = 0

� = ⇡�t > 0

�t < 0

|k±iEk± = ±|d(k)|F± = ± d

2d3

h(k) = d(k) · �

�� = �i

IC

hk� | @

@k|k�i · dk = �i

IC0h�| @

@d|�i · dd (1)

h(d) = d · �h(k, t + ⌧) = h(k)

�P =

e

2⇡

Xn2occ

ZS

FRn · dS (2)

P =

e

2⇡

Xn2occ

ZBZ

Akn dk (3)

Nn =

1

2⇡

IS

Fkn · dS (4)

FRn = rR ⇥ARn

�n =

HcAkn · dk

�n =

RSFkn · dS



1


Ion Garate

April 30, 2014



dz(k) = 0

� = 0

� = ⇡�t > 0

�t < 0

|k±iEk± = ±|d(k)|F± = ± d

2d3

h(k) = d(k) · �

�� = �i

IC

hk� | @

@k|k�i · dk = �i

IC0h�| @

@d|�i · dd (1)

h(d) = d · �h(k, t + ⌧) = h(k)

�P =

e

2⇡

Xn2occ

ZS

FRn · dS (2)

P =

e

2⇡

Xn2occ

ZBZ

Akn dk (3)

Nn =

1

2⇡

IS

Fkn · dS (4)

FRn = rR ⇥ARn

�n =

HcAkn · dk

�n =

RSFkn · dS



1


Ion Garate

April 29, 2014



dz(k) = 0

⇡ah(k) = d(k) · �h(k, t + ⌧) = h(k)

�P =

e

2⇡

Xn2occ

ZS

FRn · dS (1)

P =

e

2⇡

Xn2occ

ZBZ

Akn dk (2)

Nn =

1

2⇡

IS

Fkn · dS (3)

FRn = rR ⇥ARn

�n =

HcAkn · dk

�n =

RSFkn · dS



= gN0�

TvT · r�⇢y

(4)

2��1e↵ + W 2/D > ⌧�

�e↵ = � exp(�W 2/8l2B)

↵ = 1/2

↵ = �1/2

↵ = 3/2

↵ = 1

2��1+ W 2/D > ⌧�

�⌧� � 1

1


Ion Garate

April 29, 2014



dz(k) = 0

⇡ah(k) = d(k) · �h(k, t + ⌧) = h(k)

�P =

e

2⇡

Xn2occ

ZS

FRn · dS (1)

P =

e

2⇡

Xn2occ

ZBZ

Akn dk (2)

Nn =

1

2⇡

IS

Fkn · dS (3)

FRn = rR ⇥ARn

�n =

HcAkn · dk

�n =

RSFkn · dS



= gN0�

TvT · r�⇢y

(4)

2��1e↵ + W 2/D > ⌧�

�e↵ = � exp(�W 2/8l2B)

↵ = 1/2

↵ = �1/2

↵ = 3/2

↵ = 1

2��1+ W 2/D > ⌧�

�⌧� � 1

1


Ion Garate

April 29, 2014



dz(k) = 0

� = 0

� = ⇡�t > 0

�t < 0

h(k) = d(k) · �h(k, t + ⌧) = h(k)

�P =

e

2⇡

Xn2occ

ZS

FRn · dS (1)

P =

e

2⇡

Xn2occ

ZBZ

Akn dk (2)

Nn =

1

2⇡

IS

Fkn · dS (3)

FRn = rR ⇥ARn

�n =

HcAkn · dk

�n =

RSFkn · dS



= gN0�

TvT · r�⇢y

(4)

2��1e↵ + W 2/D > ⌧�

�e↵ = � exp(�W 2/8l2B)

↵ = 1/2

↵ = �1/2

↵ = 3/2

1


Ion Garate

April 29, 2014



dz(k) = 0

� = 0

� = ⇡�t > 0

�t < 0

h(k) = d(k) · �h(k, t + ⌧) = h(k)

�P =

e

2⇡

Xn2occ

ZS

FRn · dS (1)

P =

e

2⇡

Xn2occ

ZBZ

Akn dk (2)

Nn =

1

2⇡

IS

Fkn · dS (3)

FRn = rR ⇥ARn

�n =

HcAkn · dk

�n =

RSFkn · dS



= gN0�

TvT · r�⇢y

(4)

2��1e↵ + W 2/D > ⌧�

�e↵ = � exp(�W 2/8l2B)

↵ = 1/2

↵ = �1/2

↵ = 3/2

1


Ion Garate

April 29, 2014



dz(k) = 0

� = 0

� = ⇡�t > 0

�t < 0

h(k) = d(k) · �h(k, t + ⌧) = h(k)

�P =

e

2⇡

Xn2occ

ZS

FRn · dS (1)

P =

e

2⇡

Xn2occ

ZBZ

Akn dk (2)

Nn =

1

2⇡

IS

Fkn · dS (3)

FRn = rR ⇥ARn

�n =

HcAkn · dk

�n =

RSFkn · dS



= gN0�

TvT · r�⇢y

(4)

2��1e↵ + W 2/D > ⌧�

�e↵ = � exp(�W 2/8l2B)

↵ = 1/2

↵ = �1/2

↵ = 3/2

1


Ion Garate

April 29, 2014



dz(k) = 0

� = 0

� = ⇡�t > 0

�t < 0

h(k) = d(k) · �h(k, t + ⌧) = h(k)

�P =

e

2⇡

Xn2occ

ZS

FRn · dS (1)

P =

e

2⇡

Xn2occ

ZBZ

Akn dk (2)

Nn =

1

2⇡

IS

Fkn · dS (3)

FRn = rR ⇥ARn

�n =

HcAkn · dk

�n =

RSFkn · dS



= gN0�

TvT · r�⇢y

(4)

2��1e↵ + W 2/D > ⌧�

�e↵ = � exp(�W 2/8l2B)

↵ = 1/2

↵ = �1/2

↵ = 3/2

1

Symmetry-protected topological phases.


Ion Garate

April 30, 2014

4|�t|ARn = ihuRn|rR|uRnidx(k) = (t + �t) + (t� �t) cos(ka)


dz(k) = 0

� = 0

� = ⇡�t > 0

�t < 0

|k±iEk± = ±|d(k)|F± = ± d

2d3

h(k) = d(k) · �

�� = �i

IC

hk� | @

@k|k�i · dk = �i

IC0h�| @

@d|�i · dd (1)

h(d) = d · �h(k, t + ⌧) = h(k)

�P =

e

2⇡

Xn2occ

ZS

FRn · dS (2)

P =

e

2⇡

Xn2occ

ZBZ

Akn dk (3)

Nn =

1

2⇡

IS

Fkn · dS (4)

FRn = rR ⇥ARn

�n =

HcAkn · dk

�n =

RSFkn · dS


h(k)|ukni = Ekn|ukniH| kni = Ekn| kni

1

2 atoms per unit cell


Ion Garate

April 30, 2014

�t = 0



dz(k) = 0

� = 0

� = ⇡�t > 0

�t < 0

|k±iEk± = ±|d(k)|F± = ± d

2d3

h(k) = d(k) · �

�� = �i

IC

hk� | @

@k|k�i · dk = �i

IC0h�| @

@d|�i · dd (1)

h(d) = d · �h(k, t + ⌧) = h(k)

�P =

e

2⇡

Xn2occ

ZS

FRn · dS (2)

P =

e

2⇡

Xn2occ

ZBZ

Akn dk (3)

Nn =

1

2⇡

IS

Fkn · dS (4)

FRn = rR ⇥ARn

�n =

HcAkn · dk

�n =

RSFkn · dS



1


Ion Garate

April 29, 2014



dz(k) = 0

⇡ah(k) = d(k) · �h(k, t + ⌧) = h(k)

�P =

e

2⇡

Xn2occ

ZS

FRn · dS (1)

P =

e

2⇡

Xn2occ

ZBZ

Akn dk (2)

Nn =

1

2⇡

IS

Fkn · dS (3)

FRn = rR ⇥ARn

�n =

HcAkn · dk

�n =

RSFkn · dS



= gN0�

TvT · r�⇢y

(4)

2��1e↵ + W 2/D > ⌧�

�e↵ = � exp(�W 2/8l2B)

↵ = 1/2

↵ = �1/2

↵ = 3/2

↵ = 1

2��1+ W 2/D > ⌧�

�⌧� � 1

1


Ion Garate

April 29, 2014



dz(k) = 0

⇡ah(k) = d(k) · �h(k, t + ⌧) = h(k)

�P =

e

2⇡

Xn2occ

ZS

FRn · dS (1)

P =

e

2⇡

Xn2occ

ZBZ

Akn dk (2)

Nn =

1

2⇡

IS

Fkn · dS (3)

FRn = rR ⇥ARn

�n =

HcAkn · dk

�n =

RSFkn · dS



= gN0�

TvT · r�⇢y

(4)

2��1e↵ + W 2/D > ⌧�

�e↵ = � exp(�W 2/8l2B)

↵ = 1/2

↵ = �1/2

↵ = 3/2

↵ = 1

2��1+ W 2/D > ⌧�

�⌧� � 1

1


Ion Garate

April 29, 2014



dz(k) = 0

⇡ah(k) = d(k) · �h(k, t + ⌧) = h(k)

�P =

e

2⇡

Xn2occ

ZS

FRn · dS (1)

P =

e

2⇡

Xn2occ

ZBZ

Akn dk (2)

Nn =

1

2⇡

IS

Fkn · dS (3)

FRn = rR ⇥ARn

�n =

HcAkn · dk

�n =

RSFkn · dS



= gN0�

TvT · r�⇢y

(4)

2��1e↵ + W 2/D > ⌧�

�e↵ = � exp(�W 2/8l2B)

↵ = 1/2

↵ = �1/2

↵ = 3/2

↵ = 1

2��1+ W 2/D > ⌧�

�⌧� � 1

1


Ion Garate

April 29, 2014



dz(k) = 0

⇡ah(k) = d(k) · �h(k, t + ⌧) = h(k)

�P =

e

2⇡

Xn2occ

ZS

FRn · dS (1)

P =

e

2⇡

Xn2occ

ZBZ

Akn dk (2)

Nn =

1

2⇡

IS

Fkn · dS (3)

FRn = rR ⇥ARn

�n =

HcAkn · dk

�n =

RSFkn · dS



= gN0�

TvT · r�⇢y

(4)

2��1e↵ + W 2/D > ⌧�

�e↵ = � exp(�W 2/8l2B)

↵ = 1/2

↵ = �1/2

↵ = 3/2

↵ = 1

2��1+ W 2/D > ⌧�

�⌧� � 1

1


Ion Garate

May 4, 2014

h(k) = dx(k)�x+ dy(k)�y

Fn(R) = rR ⇥An(R)

�P =

e

2⇡

Xn2occ

IC

An(R) · dR =

e

2⇡

Xn2occ

ZS



Q =

Z ⌧

0

dt@P

@t= P (t = ⌧)� P (t = 0) ⌘ �P (2)

j = @P/@t⇢ = �@xP� = P · n14⇡

HS dS = 1� gH


nChern =

12⇡

RSFn(R) · dS


An(R) · dR =

RS

dS · [rR ⇥An(R)]


RC


1~

R t

0dt0En(R(t0))� i

RC


R(t)

�xy(B) = n e2

h sgn(B)

�h = �R� · (z ⇥ q) nc ⌘ n" + n# = 0

ns ⌘ n" � n# = 2 sgn(�so)

�sxy ⌘ �"xy � �#xy =

jx" � jx#Ey

= 2

e2

h(3)

n" = sgn(�so)

1

How does d change as k runs from


Ion Garate

May 4, 2014

�⇡/a ⇡/a h(k) = dx(k)�x+ dy(k)�y

Fn(R) = rR ⇥An(R)

�P =

e

2⇡

Xn2occ

IC

An(R) · dR =

e

2⇡

Xn2occ

ZS



Q =

Z ⌧

0

dt@P

@t= P (t = ⌧)� P (t = 0) ⌘ �P (2)

j = @P/@t⇢ = �@xP� = P · n14⇡

HS dS = 1� gH


nChern =

12⇡

RSFn(R) · dS


An(R) · dR =

RS

dS · [rR ⇥An(R)]


RC


1~

R t

0dt0En(R(t0))� i

RC


R(t)

�xy(B) = n e2

h sgn(B)

�h = �R� · (z ⇥ q) nc ⌘ n" + n# = 0

ns ⌘ n" � n# = 2 sgn(�so)

�sxy ⌘ �"xy � �#xy =

jx" � jx#Ey

= 2

e2

h(3)

n" = sgn(�so)

1

to


Ion Garate

May 4, 2014

�⇡/a ⇡/a h(k) = dx(k)�x+ dy(k)�y

Fn(R) = rR ⇥An(R)

�P =

e

2⇡

Xn2occ

IC

An(R) · dR =

e

2⇡

Xn2occ

ZS



Q =

Z ⌧

0

dt@P

@t= P (t = ⌧)� P (t = 0) ⌘ �P (2)

j = @P/@t⇢ = �@xP� = P · n14⇡

HS dS = 1� gH


nChern =

12⇡

RSFn(R) · dS


An(R) · dR =

RS

dS · [rR ⇥An(R)]


RC


1~

R t

0dt0En(R(t0))� i

RC


R(t)

�xy(B) = n e2

h sgn(B)

�h = �R� · (z ⇥ q) nc ⌘ n" + n# = 0

ns ⌘ n" � n# = 2 sgn(�so)

�sxy ⌘ �"xy � �#xy =

jx" � jx#Ey

= 2

e2

h(3)

n" = sgn(�so)

1

?

Topological quantum phase transition.

Emergence of Dirac Fermions at low energies:

E

k


Ion Garate

April 29, 2014


ah(k, t + ⌧) = h(k)

�P =

e

2⇡

ZS

FRn · dS (1)

P =

e

2⇡

Xn2occ

ZBZ

Akn dk (2)

Nn =

1

2⇡

IS

Fkn · dS (3)

FRn = rR ⇥ARn

�n =

HcAkn · dk

�n =

RSFkn · dS



= gN0�

TvT · r�⇢y

(4)

2��1e↵ + W 2/D > ⌧�

�e↵ = � exp(�W 2/8l2B)

↵ = 1/2

↵ = �1/2

↵ = 3/2

↵ = 1

2��1+ W 2/D > ⌧�

�⌧� � 1

�⌧� ⌧ 1


lB =

p~/eB||

1


Ion Garate

April 29, 2014


ah(k, t + ⌧) = h(k)

�P =

e

2⇡

ZS

FRn · dS (1)

P =

e

2⇡

Xn2occ

ZBZ

Akn dk (2)

Nn =

1

2⇡

IS

Fkn · dS (3)

FRn = rR ⇥ARn

�n =

HcAkn · dk

�n =

RSFkn · dS



= gN0�

TvT · r�⇢y

(4)

2��1e↵ + W 2/D > ⌧�

�e↵ = � exp(�W 2/8l2B)

↵ = 1/2

↵ = �1/2

↵ = 3/2

↵ = 1

2��1+ W 2/D > ⌧�

�⌧� � 1

�⌧� ⌧ 1


lB =

p~/eB||

1


Ion Garate

April 30, 2014

h(q) ' m�x+ vq�y

|�t|⌧ tk = ⇡/a� qqa ⌧ 1

m ⌘ 2�tv ⌘ at�t = 0



dz(k) = 0

� = 0

� = ⇡�t > 0

�t < 0

|k±iEk± = ±|d(k)|F± = ± d

2d3

h(k) = d(k) · �

�� = �i

IC

hk� | @

@k|k�i · dk = �i

IC0h�| @

@d|�i · dd (1)

h(d) = d · �h(k, t + ⌧) = h(k)

�P =

e

2⇡

Xn2occ

ZS

FRn · dS (2)

P =

e

2⇡

Xn2occ

ZBZ

Akn dk (3)

Nn =

1

2⇡

IS

Fkn · dS (4)

FRn = rR ⇥ARn

�n =

HcAkn · dk

1


Ion Garate

April 30, 2014


|�t|⌧ tk = ⇡/a� qqa ⌧ 1

m ⌘ 2�tv ⌘ at�t = 0



dz(k) = 0

� = 0

� = ⇡�t > 0

�t < 0

|k±iEk± = ±|d(k)|F± = ± d

2d3

h(k) = d(k) · �

�� = �i

IC

hk� | @

@k|k�i · dk = �i

IC0h�| @

@d|�i · dd (1)

h(d) = d · �h(k, t + ⌧) = h(k)

�P =

e

2⇡

Xn2occ

ZS

FRn · dS (2)

P =

e

2⇡

Xn2occ

ZBZ

Akn dk (3)

Nn =

1

2⇡

IS

Fkn · dS (4)

FRn = rR ⇥ARn

�n =

HcAkn · dk

1


Ion Garate

April 30, 2014


|�t|⌧ tk = ⇡/a� qqa ⌧ 1

m ⌘ 2�tv ⌘ at�t = 0



dz(k) = 0

� = 0

� = ⇡�t > 0

�t < 0

|k±iEk± = ±|d(k)|F± = ± d

2d3

h(k) = d(k) · �

�� = �i

IC

hk� | @

@k|k�i · dk = �i

IC0h�| @

@d|�i · dd (1)

h(d) = d · �h(k, t + ⌧) = h(k)

�P =

e

2⇡

Xn2occ

ZS

FRn · dS (2)

P =

e

2⇡

Xn2occ

ZBZ

Akn dk (3)

Nn =

1

2⇡

IS

Fkn · dS (4)

FRn = rR ⇥ARn

�n =

HcAkn · dk

1


Ion Garate

April 30, 2014


|�t|⌧ tk = ⇡/a� qqa ⌧ 1

m ⌘ 2�tv ⌘ at�t = 0



dz(k) = 0

� = 0

� = ⇡�t > 0

�t < 0

|k±iEk± = ±|d(k)|F± = ± d

2d3

h(k) = d(k) · �

�� = �i

IC

hk� | @

@k|k�i · dk = �i

IC0h�| @

@d|�i · dd (1)

h(d) = d · �h(k, t + ⌧) = h(k)

�P =

e

2⇡

Xn2occ

ZS

FRn · dS (2)

P =

e

2⇡

Xn2occ

ZBZ

Akn dk (3)

Nn =

1

2⇡

IS

Fkn · dS (4)

FRn = rR ⇥ARn

�n =

HcAkn · dk

1

1D Dirac Hamiltonian


Ion Garate

April 30, 2014


|�t|⌧ tk = ⇡/a� qqa ⌧ 1

m ⌘ 2�tv ⌘ at�t = 0



dz(k) = 0

� = 0

� = ⇡�t > 0

�t < 0

|k±iEk± = ±|d(k)|F± = ± d

2d3

h(k) = d(k) · �

�� = �i

IC

hk� | @

@k|k�i · dk = �i

IC0h�| @

@d|�i · dd (1)

h(d) = d · �h(k, t + ⌧) = h(k)

�P =

e

2⇡

Xn2occ

ZS

FRn · dS (2)

P =

e

2⇡

Xn2occ

ZBZ

Akn dk (3)

Nn =

1

2⇡

IS

Fkn · dS (4)

FRn = rR ⇥ARn

�n =

HcAkn · dk

1

Dirac mass

Velocity of Dirac fermion


Ion Garate

April 30, 2014

4|�t|ARn = ihuRn|rR|uRnidx(k) = (t + �t) + (t� �t) cos(ka)


dz(k) = 0

� = 0

� = ⇡�t > 0

�t < 0

|k±iEk± = ±|d(k)|F± = ± d

2d3

h(k) = d(k) · �

�� = �i

IC

hk� | @

@k|k�i · dk = �i

IC0h�| @

@d|�i · dd (1)

h(d) = d · �h(k, t + ⌧) = h(k)

�P =

e

2⇡

Xn2occ

ZS

FRn · dS (2)

P =

e

2⇡

Xn2occ

ZBZ

Akn dk (3)

Nn =

1

2⇡

IS

Fkn · dS (4)

FRn = rR ⇥ARn

�n =

HcAkn · dk

�n =

RSFkn · dS



1

E

2m0 Zero energy state

m(z)

z-m0

m0

Jackiw-Rebbi zero mode: localized at domain wall, decay length

Insensitive to details of the Hamiltonian.


Ion Garate

April 29, 2014



dz(k) = 0

� = 0

� = ⇡�t > 0

�t < 0

h(k) = d(k) · �h(k, t + ⌧) = h(k)

�P =

e

2⇡

Xn2occ

ZS

FRn · dS (1)

P =

e

2⇡

Xn2occ

ZBZ

Akn dk (2)

Nn =

1

2⇡

IS

Fkn · dS (3)

FRn = rR ⇥ARn

�n =

HcAkn · dk

�n =

RSFkn · dS



= gN0�

TvT · r�⇢y

(4)

2��1e↵ + W 2/D > ⌧�

�e↵ = � exp(�W 2/8l2B)

↵ = 1/2

↵ = �1/2

↵ = 3/2

1


Ion Garate

April 29, 2014



dz(k) = 0

� = 0

� = ⇡�t > 0

�t < 0

h(k) = d(k) · �h(k, t + ⌧) = h(k)

�P =

e

2⇡

Xn2occ

ZS

FRn · dS (1)

P =

e

2⇡

Xn2occ

ZBZ

Akn dk (2)

Nn =

1

2⇡

IS

Fkn · dS (3)

FRn = rR ⇥ARn

�n =

HcAkn · dk

�n =

RSFkn · dS



= gN0�

TvT · r�⇢y

(4)

2��1e↵ + W 2/D > ⌧�

�e↵ = � exp(�W 2/8l2B)

↵ = 1/2

↵ = �1/2

↵ = 3/2

1

Emergence of zero modes at “domain walls”


Ion Garate

May 1, 2014

v/m0

h(z) ' m(z)�x+�iv�y@x

|�t|⌧ tk = ⇡/a� qqa ⌧ 1

m ⌘ 2�tv ⌘ at�t = 0



dz(k) = 0

� = 0

� = ⇡�t > 0

�t < 0

|k±iEk± = ±|d(k)|F± = ± d

2d3

h(k) = d(k) · �

�� = �i

IC

hk� | @

@k|k�i · dk = �i

IC0h�| @

@d|�i · dd (1)

h(d) = d · �h(k, t + ⌧) = h(k)

�P =

e

2⇡

Xn2occ

ZS

FRn · dS (2)

P =

e

2⇡

Xn2occ

ZBZ

Akn dk (3)

Nn =

1

2⇡

IS

Fkn · dS (4)

FRn = rR ⇥ARn

1

mH = +�so

mH = ��so

�xy =

e2

h sgn(mH)

mK = mK0

nc = nK + nK0= 0

mK = �mK0


|q�i|q+i

⇧(q,!) =

1

V

Xk

Xnn0


Ekn � Ek�qn0 � !(4)

�Ekn 'Xn0k0

g2q


(5)

⇢q =

Xnn0

Xk,k0


�h = mS�z

�h = mH�z⌧z


n� =

12⇡

R1 dS · Fq� =

12 sgn(mvxvy)


�xx = �yy = 0

�xy =

e2

h

Xn2occ

1

2⇡

ZBZ

dS · Fkn (7)

�fkn

�| kni

j = eXkn

h kn|v| knifkn (8)

En = !c(n + 1/2)

H =

12m

hp2

x +

�py � eB

c x�2

iv/m0

h(z) ' m(z)�x � iv�y@z

|�t| ⌧ tk = ⇡/a� qqa ⌧ 1

m ⌘ 2�tv ⌘ at�t = 0

ARn = ihuRn|rR|uRni

2

This result can be generalized to 2 and 3 dimensions.

Band Topology in 2D

Quantum Hall Insulator

B


Ion Garate

May 1, 2014

H =

12m

hp2

x +

�py � eB

c x�2

iv/m0


|�t|⌧ tk = ⇡/a� qqa ⌧ 1

m ⌘ 2�tv ⌘ at�t = 0



dz(k) = 0

� = 0

� = ⇡�t > 0

�t < 0

|k±iEk± = ±|d(k)|F± = ± d

2d3

h(k) = d(k) · �

�� = �i

IC

hk� | @

@k|k�i · dk = �i

IC0h�| @

@d|�i · dd (1)

h(d) = d · �h(k, t + ⌧) = h(k)

�P =

e

2⇡

Xn2occ

ZS

FRn · dS (2)

P =

e

2⇡

Xn2occ

ZBZ

Akn dk (3)

Nn =

1

2⇡

IS

Fkn · dS (4)

1

z

x y

(Landau gauge)

E

k


Ion Garate

May 1, 2014

En = !c(n + 1/2)

H =

12m

hp2

x +

�py � eB

c x�2

iv/m0


|�t|⌧ tk = ⇡/a� qqa ⌧ 1

m ⌘ 2�tv ⌘ at�t = 0



dz(k) = 0

� = 0

� = ⇡�t > 0

�t < 0

|k±iEk± = ±|d(k)|F± = ± d

2d3

h(k) = d(k) · �

�� = �i

IC

hk� | @

@k|k�i · dk = �i

IC0h�| @

@d|�i · dd (1)

h(d) = d · �h(k, t + ⌧) = h(k)

�P =

e

2⇡

Xn2occ

ZS

FRn · dS (2)

P =

e

2⇡

Xn2occ

ZBZ

Akn dk (3)

Nn =

1

2⇡

IS

Fkn · dS (4)

1

is a good quantum number.

mH = +�so

mH = ��so

�xy =

e2

h sgn(mH)

mK = mK0

nc = nK + nK0= 0

mK = �mK0


|q�i|q+i

⇧(q,!) =

1

V

Xk

Xnn0


Ekn � Ek�qn0 � !(4)

�Ekn 'Xn0k0

g2q


(5)

⇢q =

Xnn0

Xk,k0


�h = mS�z

�h = mH�z⌧z


n� =

12⇡

R1 dS · Fq� =

12 sgn(mvxvy)


�xx = �yy = 0

�xy =

e2

h

Xn2occ

1

2⇡

ZBZ

dS · Fkn (7)

�fkn

�| kni

j = eXkn

h kn|v| knifkn (8)

En = !c(n + 1/2)

H =

12m

hp2

x +

�py � eB

c x�2

iv/m0


|�t| ⌧ tk = ⇡/a� qqa ⌧ 1

m ⌘ 2�tv ⌘ at�t = 0

ARn = ihuRn|rR|uRni

2

In the x-direction, a harmonic oscillator with frequency centered at


Ion Garate

May 4, 2014

c py/(eB)

�⇡/a ⇡/a h(k) = dx(k)�x+ dy(k)�y

Fn(R) = rR ⇥An(R)

�P =

e

2⇡

Xn2occ

IC

An(R) · dR =

e

2⇡

Xn2occ

ZS



Q =

Z ⌧

0

dt@P

@t= P (t = ⌧)� P (t = 0) ⌘ �P (2)

j = @P/@t⇢ = �@xP� = P · n14⇡

HS dS = 1� gH


nChern =

12⇡

RSFn(R) · dS


An(R) · dR =

RS

dS · [rR ⇥An(R)]


RC


1~

R t

0dt0En(R(t0))� i

RC


R(t)

�xy(B) = n e2

h sgn(B)

�h = �R� · (z ⇥ q) nc ⌘ n" + n# = 0

ns ⌘ n" � n# = 2 sgn(�so)

�sxy ⌘ �"xy � �#xy =

jx" � jx#Ey

= 2

e2

h(3)

n" = sgn(�so)

1

mH = +�so

mH = ��so

�xy =

e2

h sgn(mH)

mK = mK0

nc = nK + nK0= 0

mK = �mK0


|q�i|q+i

⇧(q,!) =

1

V

Xk

Xnn0


Ekn � Ek�qn0 � !(4)

�Ekn 'Xn0k0

g2q


(5)

⇢q =

Xnn0

Xk,k0


�h = mS�z

�h = mH�z⌧z


n� =

12⇡

R1 dS · Fq� =

12 sgn(mvxvy)


�xx = �yy = 0

�xy =

e2

h

Xn2occ

1

2⇡

ZBZ

dS · Fkn (7)

�fkn

�| kni

j = eXkn

h kn|v| knifkn (8)

En = !c(n + 1/2)

H =

12m

hp2

x +

�py � eB

c x�2

iv/m0


|�t| ⌧ tk = ⇡/a� qqa ⌧ 1

m ⌘ 2�tv ⌘ at�t = 0

ARn = ihuRn|rR|uRni

2


Ion Garate

May 4, 2014

!c = eB/(mc)c py/(eB)

�⇡/a ⇡/a h(k) = dx(k)�x+ dy(k)�y

Fn(R) = rR ⇥An(R)

�P =

e

2⇡

Xn2occ

IC

An(R) · dR =

e

2⇡

Xn2occ

ZS



Q =

Z ⌧

0

dt@P

@t= P (t = ⌧)� P (t = 0) ⌘ �P (2)

j = @P/@t⇢ = �@xP� = P · n14⇡

HS dS = 1� gH


nChern =

12⇡

RSFn(R) · dS


An(R) · dR =

RS

dS · [rR ⇥An(R)]


RC


1~

R t

0dt0En(R(t0))� i

RC


R(t)

�xy(B) = n e2

h sgn(B)

�h = �R� · (z ⇥ q) nc ⌘ n" + n# = 0

ns ⌘ n" � n# = 2 sgn(�so)

�sxy ⌘ �"xy � �#xy =

jx" � jx#Ey

= 2

e2

h(3)

1

The Hall conductivity of a 2D band insulator is always quantized


Ion Garate

May 1, 2014

j = eXkn

h kn|v| knifkn (1)

En = !c(n + 1/2)

H =

12m

hp2

x +

�py � eB

c x�2

iv/m0


|�t|⌧ tk = ⇡/a� qqa ⌧ 1

m ⌘ 2�tv ⌘ at�t = 0



dz(k) = 0

� = 0

� = ⇡�t > 0

�t < 0

|k±iEk± = ±|d(k)|F± = ± d

2d3

h(k) = d(k) · �

�� = �i

IC

hk� | @@k

|k�i · dk = �i

IC0h�| @

@d|�i · dd (2)

h(d) = d · �h(k, t + ⌧) = h(k)

�P =

e

2⇡

Xn2occ

ZS

FRn · dS (3)

1


Ion Garate

May 1, 2014

�fkn

�| kni

j = eXkn

h kn|v| knifkn (1)

En = !c(n + 1/2)

H =

12m

hp2

x +

�py � eB

c x�2

iv/m0


|�t|⌧ tk = ⇡/a� qqa ⌧ 1

m ⌘ 2�tv ⌘ at�t = 0



dz(k) = 0

� = 0

� = ⇡�t > 0

�t < 0

|k±iEk± = ±|d(k)|F± = ± d

2d3

h(k) = d(k) · �

�� = �i

IC

hk� | @@k

|k�i · dk = �i

IC0h�| @

@d|�i · dd (2)

h(d) = d · �h(k, t + ⌧) = h(k)

�P =

e

2⇡

Xn2occ

ZS

FRn · dS (3)

1


Ion Garate

May 1, 2014

�fkn

�| kni

j = eXkn

h kn|v| knifkn (1)

En = !c(n + 1/2)

H =

12m

hp2

x +

�py � eB

c x�2

iv/m0


|�t|⌧ tk = ⇡/a� qqa ⌧ 1

m ⌘ 2�tv ⌘ at�t = 0



dz(k) = 0

� = 0

� = ⇡�t > 0

�t < 0

|k±iEk± = ±|d(k)|F± = ± d

2d3

h(k) = d(k) · �

�� = �i

IC

hk� | @@k

|k�i · dk = �i

IC0h�| @

@d|�i · dd (2)

h(d) = d · �h(k, t + ⌧) = h(k)

�P =

e

2⇡

Xn2occ

ZS

FRn · dS (3)

1

Vanishes in equilibrium.

An applied electric field can induce a current in two ways:

(i) By changing the population of the states:

E

k


Ion Garate

April 29, 2014


ah(k, t + ⌧) = h(k)

�P =

e

2⇡

ZS

FRn · dS (1)

P =

e

2⇡

Xn2occ

ZBZ

Akn dk (2)

Nn =

1

2⇡

IS

Fkn · dS (3)

FRn = rR ⇥ARn

�n =

HcAkn · dk

�n =

RSFkn · dS



= gN0�

TvT · r�⇢y

(4)

2��1e↵ + W 2/D > ⌧�

�e↵ = � exp(�W 2/8l2B)

↵ = 1/2

↵ = �1/2

↵ = 3/2

↵ = 1

2��1+ W 2/D > ⌧�

�⌧� � 1

�⌧� ⌧ 1


lB =

p~/eB||

1


Ion Garate

April 29, 2014


ah(k, t + ⌧) = h(k)

�P =

e

2⇡

ZS

FRn · dS (1)

P =

e

2⇡

Xn2occ

ZBZ

Akn dk (2)

Nn =

1

2⇡

IS

Fkn · dS (3)

FRn = rR ⇥ARn

�n =

HcAkn · dk

�n =

RSFkn · dS



= gN0�

TvT · r�⇢y

(4)

2��1e↵ + W 2/D > ⌧�

�e↵ = � exp(�W 2/8l2B)

↵ = 1/2

↵ = �1/2

↵ = 3/2

↵ = 1

2��1+ W 2/D > ⌧�

�⌧� � 1

�⌧� ⌧ 1


lB =

p~/eB||

1

(ii) By changing the eigenstates:

ky

kx


Ion Garate

April 29, 2014


ah(k, t + ⌧) = h(k)

�P =

e

2⇡

ZS

FRn · dS (1)

P =

e

2⇡

Xn2occ

ZBZ

Akn dk (2)

Nn =

1

2⇡

IS

Fkn · dS (3)

FRn = rR ⇥ARn

�n =

HcAkn · dk

�n =

RSFkn · dS



= gN0�

TvT · r�⇢y

(4)

2��1e↵ + W 2/D > ⌧�

�e↵ = � exp(�W 2/8l2B)

↵ = 1/2

↵ = �1/2

↵ = 3/2

↵ = 1

2��1+ W 2/D > ⌧�

�⌧� � 1

�⌧� ⌧ 1


lB =

p~/eB||

1


Ion Garate

April 29, 2014


ah(k, t + ⌧) = h(k)

�P =

e

2⇡

ZS

FRn · dS (1)

P =

e

2⇡

Xn2occ

ZBZ

Akn dk (2)

Nn =

1

2⇡

IS

Fkn · dS (3)

FRn = rR ⇥ARn

�n =

HcAkn · dk

�n =

RSFkn · dS



= gN0�

TvT · r�⇢y

(4)

2��1e↵ + W 2/D > ⌧�

�e↵ = � exp(�W 2/8l2B)

↵ = 1/2

↵ = �1/2

↵ = 3/2

↵ = 1

2��1+ W 2/D > ⌧�

�⌧� � 1

�⌧� ⌧ 1


lB =

p~/eB||

1


Ion Garate

April 29, 2014


�P =

e

2⇡

ZS

FRn · dS (1)

P =

e

2⇡

Xn2occ

ZBZ

Akn dk (2)

Nn =

1

2⇡

IS

Fkn · dS (3)

FRn = rR ⇥ARn

�n =

HcAkn · dk

�n =

RSFkn · dS



= gN0�

TvT · r�⇢y

(4)

2��1e↵ + W 2/D > ⌧�

�e↵ = � exp(�W 2/8l2B)

↵ = 1/2

↵ = �1/2

↵ = 3/2

↵ = 1

2��1+ W 2/D > ⌧�

�⌧� � 1

�⌧� ⌧ 1


lB =

p~/eB||

1

Chern number

Electric current in a crystal.

n" = sgn(�so)

mH = +�so

mH = ��so

�xy =

e2

h sgn(mH)

mK = mK0

nc = nK + nK0= 0

mK = �mK0


|q�i|q+i

⇧(q,!) =

1

V

Xk

Xnn0


Ekn � Ek�qn0 � !(4)

�Ekn 'Xn0k0

g2q


(5)

⇢q =

Xnn0

Xk,k0


�h = mS�z

�h = mH�z⌧z


n� =

12⇡

R1 dS · Fq� =

12 sgn(mvxvy)


�xx = �yy = 0

�xy =

e2

h

Xn2occ

1

2⇡

ZS

dS · Fn(k) (7)

�fkn

�| kni

j = eXkn

h kn|v| knifkn (8)

En = !c(n + 1/2)

H =

12m

hp2

x +

�py � eB

c x�2

iv/m0


|�t| ⌧ tk = ⇡/a� qqa ⌧ 1

m ⌘ 2�tv ⌘ at�t = 0

2

S

Chern number vanishes in presence of time-reversal symmetry

Edge states in quantum Hall insulators:

Robustness against backscattering

Electrons on same edge move along the same direction.

# of edge states at the Fermi level= # of occupied bulk LLs =total Chern number of occupied LLs

LLs bend near sample edge.

B z

x y Assume infinite length along y,

finite length along x.

Fermi level intersects LLs at the edge.

E

x

Fermi level.

Electrons on opposite edges move along the opposite directions.

Graphene Graphene E

k

Two band model Novoselov et al. ‘05

www.univie.ac.at

( ) ( )H k d k( ) | ( ) |E k d k

ˆ ( , )x yk kd

Inversion and Time reversal symmetry require ( ) 0zd k2D Dirac points at : point zeros in ( , )x yd d

( ) vH K q q Massless Dirac Hamiltonian

k K-K +K

Berry’s phase around Dirac point

AB

†Ai Bj

ij

H t c c


Ion Garate

April 30, 2014



dz(k) = 0

� = 0

� = ⇡�t > 0

�t < 0

F =

d2d3

h(k) = d(k) · �

�� =

IC

ihk� |rk|k�i · dk (1)

h(d) = d · �h(k, t + ⌧) = h(k)

�P =

e

2⇡

Xn2occ

ZS

FRn · dS (2)

P =

e

2⇡

Xn2occ

ZBZ

Akn dk (3)

Nn =

1

2⇡

IS

Fkn · dS (4)

FRn = rR ⇥ARn

�n =

HcAkn · dk

�n =

RSFkn · dS



= gN0�

TvT · r�⇢y

(5)

1

One orbital per site

No spin (for now)

Two atoms per unit cell (A and B)

Emergence of massless Dirac fermions at low energies:

Due to time-reversal symmetry and inversion symmetry, dz=0


Ion Garate

May 1, 2014

h(q) = v⌧z�xqx + v�yqy

dz(k) = 0

�xx = �yy = 0

�xy =

e2

h

Xn2occ

1

2⇡

ZBZ

dS · Fkn (1)

�fkn

�| kni

j = eXkn

h kn|v| knifkn (2)

En = !c(n + 1/2)

H =

12m

hp2

x +

�py � eB

c x�2

iv/m0


|�t|⌧ tk = ⇡/a� qqa ⌧ 1

m ⌘ 2�tv ⌘ at�t = 0



dz(k) = 0

� = 0

� = ⇡�t > 0

�t < 0

|k±iEk± = ±|d(k)|F± = ± d

2d3

h(k) = d(k) · �

1

A/B pseudospin

K/K’ pseudospin

Momentum measured from Dirac node

How to make spinless graphene insulating

Need to break either time-reversal symmetry or inversion symmetry

(i) Break inversion symmetry

E

q

(ii) Break time-reversal symmetry


Ion Garate

May 4, 2014

h(q) = v ⌧z�x qx + v �yqy + dz(q)�z


�⇡/a ⇡/a h(k) = dx(k)�x+ dy(k)�y

Fn(R) = rR ⇥An(R)

�P =

e

2⇡

Xn2occ

IC

An(R) · dR =

e

2⇡

Xn2occ

ZS



Q =

Z ⌧

0

dt@P

@t= P (t = ⌧)� P (t = 0) ⌘ �P (2)

j = @P/@t⇢ = �@xP� = P · n14⇡

HS dS = 1� gH


nChern =

12⇡

RSFn(R) · dS


An(R) · dR =

RS

dS · [rR ⇥An(R)]


RC


1~

R t

0dt0En(R(t0))� i

RC


R(t)

�xy(B) = n e2

h sgn(B)

�h = �R� · (z ⇥ q) nc ⌘ n" + n# = 0

1


Ion Garate

May 4, 2014



�⇡/a ⇡/a h(k) = dx(k)�x+ dy(k)�y

Fn(R) = rR ⇥An(R)

�P =

e

2⇡

Xn2occ

IC

An(R) · dR =

e

2⇡

Xn2occ

ZS



Q =

Z ⌧

0

dt@P

@t= P (t = ⌧)� P (t = 0) ⌘ �P (2)

j = @P/@t⇢ = �@xP� = P · n14⇡

HS dS = 1� gH


nChern =

12⇡

RSFn(R) · dS


An(R) · dR =

RS

dS · [rR ⇥An(R)]


RC


1~

R t

0dt0En(R(t0))� i

RC


R(t)

�xy(B) = n e2

h sgn(B)

1


Ion Garate

May 4, 2014




�⇡/a ⇡/a h(k) = dx(k)�x+ dy(k)�y

Fn(R) = rR ⇥An(R)

�P =

e

2⇡

Xn2occ

IC

An(R) · dR =

e

2⇡

Xn2occ

ZS



Q =

Z ⌧

0

dt@P

@t= P (t = ⌧)� P (t = 0) ⌘ �P (2)

j = @P/@t⇢ = �@xP� = P · n14⇡

HS dS = 1� gH


nChern =

12⇡

RSFn(R) · dS


An(R) · dR =

RS

dS · [rR ⇥An(R)]


RC


1~

R t

0dt0En(R(t0))� i

RC


R(t)

1

Semenoff insulator (1984)


Ion Garate

May 4, 2014




�⇡/a ⇡/a h(k) = dx(k)�x+ dy(k)�y

Fn(R) = rR ⇥An(R)

�P =

e

2⇡

Xn2occ

IC

An(R) · dR =

e

2⇡

Xn2occ

ZS



Q =

Z ⌧

0

dt@P

@t= P (t = ⌧)� P (t = 0) ⌘ �P (2)

j = @P/@t⇢ = �@xP� = P · n14⇡

HS dS = 1� gH


nChern =

12⇡

RSFn(R) · dS


An(R) · dR =

RS

dS · [rR ⇥An(R)]


RC


1~

R t

0dt0En(R(t0))� i

RC


R(t)

1

Haldane insulator (1988) [a.k.a. Chern insulator]

Chern number in spinless graphene

Semenoff insulator:

Haldane/Chern insulator:


Ion Garate

May 2, 2014

mK = mK0

nc = nK + nK0= 0

mK = �mK0


|q�i|q+i

⇧(q,!) =

1

V

Xk

Xnn0


Ekn � Ek�qn0 � !(1)

�Ekn 'Xn0k0

g2q


(2)

⇢q =

Xnn0

Xk,k0


�h = mS�z

�h = mH�z⌧z

h(q) = vx�xqx + vy�yqy + m�z

n� =

12⇡

R1 dS · Fq� =

12 sgn(mvxvy)


�xx = �yy = 0

�xy =

e2

h

Xn2occ

1

2⇡

ZBZ

dS · Fkn (4)

�fkn

�| kni

j = eXkn

h kn|v| knifkn (5)

En = !c(n + 1/2)

H =

12m

hp2

x +

�py � eB

c x�2

iv/m0

1


Ion Garate

May 2, 2014

mK = mK0

nc = nK + nK0= 0

mK = �mK0


|q�i|q+i

⇧(q,!) =

1

V

Xk

Xnn0


Ekn � Ek�qn0 � !(1)

�Ekn 'Xn0k0

g2q


(2)

⇢q =

Xnn0

Xk,k0


�h = mS�z

�h = mH�z⌧z


n� =

12⇡

R1 dS · Fq� =

12 sgn(mvxvy)


�xx = �yy = 0

�xy =

e2

h

Xn2occ

1

2⇡

ZBZ

dS · Fkn (4)

�fkn

�| kni

j = eXkn

h kn|v| knifkn (5)

En = !c(n + 1/2)

H =

12m

hp2

x +

�py � eB

c x�2

iv/m0

1

k


Ion Garate

May 2, 2014

�xy =

e2

h sgn(mH)

mK = mK0

nc = nK + nK0= 0

mK = �mK0


|q�i|q+i

⇧(q,!) =

1

V

Xk

Xnn0


Ekn � Ek�qn0 � !(1)

�Ekn 'Xn0k0

g2q


(2)

⇢q =

Xnn0

Xk,k0


�h = mS�z

�h = mH�z⌧z


n� =

12⇡

R1 dS · Fq� =

12 sgn(mvxvy)


�xx = �yy = 0

�xy =

e2

h

Xn2occ

1

2⇡

ZBZ

dS · Fkn (4)

�fkn

�| kni

j = eXkn

h kn|v| knifkn (5)

En = !c(n + 1/2)

H =

12m

hp2

x +

�py � eB

c x�2

i

1


Ion Garate

May 2, 2014

�xy =

e2

h sgn(mH)

mK = mK0

nc = nK + nK0= 0

mK = �mK0


|q�i|q+i

⇧(q,!) =

1

V

Xk

Xnn0


Ekn � Ek�qn0 � !(1)

�Ekn 'Xn0k0

g2q


(2)

⇢q =

Xnn0

Xk,k0


�h = mS�z

�h = mH�z⌧z


n� =

12⇡

R1 dS · Fq� =

12 sgn(mvxvy)


�xx = �yy = 0

�xy =

e2

h

Xn2occ

1

2⇡

ZBZ

dS · Fkn (4)

�fkn

�| kni

j = eXkn

h kn|v| knifkn (5)

En = !c(n + 1/2)

H =

12m

hp2

x +

�py � eB

c x�2

i

1


Ion Garate

May 4, 2014

|+i|�idz(q) = mS dz(q) = mH⌧z



�⇡/a ⇡/a h(k) = dx(k)�x+ dy(k)�y

Fn(R) = rR ⇥An(R)

�P =

e

2⇡

Xn2occ

IC

An(R) · dR =

e

2⇡

Xn2occ

ZS



Q =

Z ⌧

0

dt@P

@t= P (t = ⌧)� P (t = 0) ⌘ �P (2)

j = @P/@t⇢ = �@xP� = P · n14⇡

HS dS = 1� gH


nChern =

12⇡

RSFn(R) · dS


An(R) · dR =

RS

dS · [rR ⇥An(R)]


RC

An(R) · dRH(R)|n,Ri = En(R)|n,Ri| (t = 0)i = |n,R(0)i| (t)i = ei✓(t)|n,R(t)i

1


Ion Garate

May 4, 2014




�⇡/a ⇡/a h(k) = dx(k)�x+ dy(k)�y

Fn(R) = rR ⇥An(R)

�P =

e

2⇡

Xn2occ

IC

An(R) · dR =

e

2⇡

Xn2occ

ZS



Q =

Z ⌧

0

dt@P

@t= P (t = ⌧)� P (t = 0) ⌘ �P (2)

j = @P/@t⇢ = �@xP� = P · n14⇡

HS dS = 1� gH


nChern =

12⇡

RSFn(R) · dS


An(R) · dR =

RS

dS · [rR ⇥An(R)]


RC


1


Ion Garate

May 4, 2014

|+i|�inK =

12 sgn(vx

KvyKmK)




�⇡/a ⇡/a h(k) = dx(k)�x+ dy(k)�y

Fn(R) = rR ⇥An(R)

�P =

e

2⇡

Xn2occ

IC

An(R) · dR =

e

2⇡

Xn2occ

ZS



Q =

Z ⌧

0

dt@P

@t= P (t = ⌧)� P (t = 0) ⌘ �P (2)

j = @P/@t⇢ = �@xP� = P · n14⇡

HS dS = 1� gH


nChern =

12⇡

RSFn(R) · dS


An(R) · dR =

RS

dS · [rR ⇥An(R)]


RC

An(R) · dRH(R)|n,Ri = En(R)|n,Ri| (t = 0)i = |n,R(0)i

1

(similarly for K’)

“Partial” Chern number for Dirac fermion at K:

Total Chern # of a band:


Ion Garate

May 27, 2014

nChern = nK + nK0

nChern = 0

nChern = sgn(mH)

he↵(k) ⌘ �G�1(0,k) = h(k) + ⌃(0,k)

@|E⇤g |/@T

@!⇤q/@|Eg| > 0

@!⇤q/@|Eg| < 0

g2q |h0n|qn0i|2

�E0n =

Xn0q

g2q|hn0|n0qi|2E0n � Eqn0

(1)

E⇤g 6= 2m⇤

Eg = 2mAn(R) = ihn,R|rR|n,Ri� = 0

� = 1

d|Eg|/dT < 0

d|Eg|/dT > 0

0nkn0

�so < 1mK

nK = nK0

nK = �nK0


|+i|�inK =

12 sgn(vx

KvyKmK)



h(d) = d · �|±iE± = ±|d|!c = eB/(mc)

1


Ion Garate

May 27, 2014

nChern = nK + nK0

nChern = 0

nChern = sgn(mH)

he↵(k) ⌘ �G�1(0,k) = h(k) + ⌃(0,k)

@|E⇤g |/@T

@!⇤q/@|Eg| > 0

@!⇤q/@|Eg| < 0

g2q |h0n|qn0i|2

�E0n =

Xn0q


(1)

E⇤g 6= 2m⇤


� = 1

d|Eg|/dT < 0

d|Eg|/dT > 0

0nkn0

�so < 1mK

nK = nK0

nK = �nK0


|+i|�inK =

12 sgn(vx

KvyKmK)



h(d) = d · �|±iE± = ±|d|!c = eB/(mc)

1


Ion Garate

May 27, 2014

nChern = nK + nK0

nChern = 0

nChern = sgn(mH)

he↵(k) ⌘ �G�1(0,k) = h(k) + ⌃(0,k)

@|E⇤g |/@T

@!⇤q/@|Eg| > 0

@!⇤q/@|Eg| < 0

g2q |h0n|qn0i|2

�E0n =

Xn0q


(1)

E⇤g 6= 2m⇤


� = 1

d|Eg|/dT < 0

d|Eg|/dT > 0

0nkn0

�so < 1mK

nK = nK0

nK = �nK0


|+i|�inK =

12 sgn(vx

KvyKmK)



h(d) = d · �|±iE± = ±|d|!c = eB/(mc)

1

Trivial insulator

Topological insulator

Edge states m(z)

zVacuum

Chern insulator


Ion Garate

May 2, 2014

�xy =

e2

h sgn(mH)

mK = mK0

nc = nK + nK0= 0

mK = �mK0


|q�i|q+i

⇧(q,!) =

1

V

Xk

Xnn0


Ekn � Ek�qn0 � !(1)

�Ekn 'Xn0k0

g2q


(2)

⇢q =

Xnn0

Xk,k0


�h = mS�z

�h = mH�z⌧z


n� =

12⇡

R1 dS · Fq� =

12 sgn(mvxvy)


�xx = �yy = 0

�xy =

e2

h

Xn2occ

1

2⇡

ZBZ

dS · Fkn (4)

�fkn

�| kni

j = eXkn

h kn|v| knifkn (5)

En = !c(n + 1/2)

H =

12m

hp2

x +

�py � eB

c x�2

i

1


Ion Garate

May 2, 2014

�xy =

e2

h sgn(mH)

mK = mK0

nc = nK + nK0= 0

mK = �mK0


|q�i|q+i

⇧(q,!) =

1

V

Xk

Xnn0


Ekn � Ek�qn0 � !(1)

�Ekn 'Xn0k0

g2q


(2)

⇢q =

Xnn0

Xk,k0


�h = mS�z

�h = mH�z⌧z


n� =

12⇡

R1 dS · Fq� =

12 sgn(mvxvy)


�xx = �yy = 0

�xy =

e2

h

Xn2occ

1

2⇡

ZBZ

dS · Fkn (4)

�fkn

�| kni

j = eXkn

h kn|v| knifkn (5)

En = !c(n + 1/2)

H =

12m

hp2

x +

�py � eB

c x�2

i

1

k


Ion Garate

May 2, 2014

�xy =

e2

h sgn(mH)

mK = mK0

nc = nK + nK0= 0

mK = �mK0


|q�i|q+i

⇧(q,!) =

1

V

Xk

Xnn0


Ekn � Ek�qn0 � !(1)

�Ekn 'Xn0k0

g2q


(2)

⇢q =

Xnn0

Xk,k0


�h = mS�z

�h = mH�z⌧z


n� =

12⇡

R1 dS · Fq� =

12 sgn(mvxvy)


�xx = �yy = 0

�xy =

e2

h

Xn2occ

1

2⇡

ZBZ

dS · Fkn (4)

�fkn

�| kni

j = eXkn

h kn|v| knifkn (5)

En = !c(n + 1/2)

H =

12m

hp2

x +

�py � eB

c x�2

i

1


Ion Garate

May 2, 2014

�xy =

e2

h sgn(mH)

mK = mK0

nc = nK + nK0= 0

mK = �mK0


|q�i|q+i

⇧(q,!) =

1

V

Xk

Xnn0


Ekn � Ek�qn0 � !(1)

�Ekn 'Xn0k0

g2q


(2)

⇢q =

Xnn0

Xk,k0


�h = mS�z

�h = mH�z⌧z


n� =

12⇡

R1 dS · Fq� =

12 sgn(mvxvy)


�xx = �yy = 0

�xy =

e2

h

Xn2occ

1

2⇡

ZBZ

dS · Fkn (4)

�fkn

�| kni

j = eXkn

h kn|v| knifkn (5)

En = !c(n + 1/2)

H =

12m

hp2

x +

�py � eB

c x�2

i

1


Ion Garate

May 2, 2014

�xy =

e2

h sgn(mH)

mK = mK0

nc = nK + nK0= 0

mK = �mK0


|q�i|q+i

⇧(q,!) =

1

V

Xk

Xnn0


Ekn � Ek�qn0 � !(1)

�Ekn 'Xn0k0

g2q


(2)

⇢q =

Xnn0

Xk,k0


�h = mS�z

�h = mH�z⌧z


n� =

12⇡

R1 dS · Fq� =

12 sgn(mvxvy)


�xx = �yy = 0

�xy =

e2

h

Xn2occ

1

2⇡

ZBZ

dS · Fkn (4)

�fkn

�| kni

j = eXkn

h kn|v| knifkn (5)

En = !c(n + 1/2)

H =

12m

hp2

x +

�py � eB

c x�2

i

1

Propagating mode localized along edge

Quantum Hall effect without magnetic field (1988)

Kane-Mele model (2005) Graphene E

k

Two band model Novoselov et al. ‘05

www.univie.ac.at

( ) ( )H k d k( ) | ( ) |E k d k

ˆ ( , )x yk kd

Inversion and Time reversal symmetry require ( ) 0zd k2D Dirac points at : point zeros in ( , )x yd d

( ) vH K q q Massless Dirac Hamiltonian

k K-K +K

Berry’s phase around Dirac point

AB

†Ai Bj

ij

H t c c

One orbital per site

Include spin

Two atoms per unit cell (A and B)

Opens a gap at K and K’ without breaking any symmetries

spin

Spin-degenerate energy spectrum.

Low-energy effective Hamiltonian:


Ion Garate

May 2, 2014

�xy =

e2

h sgn(mH)

mK = mK0

nc = nK + nK0= 0

mK = �mK0


|q�i|q+i

⇧(q,!) =

1

V

Xk

Xnn0


Ekn � Ek�qn0 � !(1)

�Ekn 'Xn0k0

g2q


(2)

⇢q =

Xnn0

Xk,k0


�h = mS�z

�h = mH�z⌧z


n� =

12⇡

R1 dS · Fq� =

12 sgn(mvxvy)


�xx = �yy = 0

�xy =

e2

h

Xn2occ

1

2⇡

ZBZ

dS · Fkn (4)

�fkn

�| kni

j = eXkn

h kn|v| knifkn (5)

En = !c(n + 1/2)

H =

12m

hp2

x +

�py � eB

c x�2

i

1

sz is conserved.


Ion Garate

May 2, 2014

nc ⌘ n" + n# = 0

ns ⌘ n" � n# = 2 sgn(�so)

�sxy =

jx" � jx#Ey

= �"xy � �#xy = 2

e2

h(1)

n" = sgn(�so)

mH = +�so

mH = ��so

�xy =

e2

h sgn(mH)

mK = mK0

nc = nK + nK0= 0

mK = �mK0


|q�i|q+i

⇧(q,!) =

1

V

Xk

Xnn0


Ekn � Ek�qn0 � !(2)

�Ekn 'Xn0k0

g2q


(3)

⇢q =

Xnn0

Xk,k0


�h = mS�z

�h = mH�z⌧z


n� =

12⇡

R1 dS · Fq� =

12 sgn(mvxvy)


�xx = �yy = 0

�xy =

e2

h

Xn2occ

1

2⇡

ZBZ

dS · Fkn (5)

1


Ion Garate

May 2, 2014

nc ⌘ n" + n# = 0

ns ⌘ n" � n# = 2sgn(�so)

n" = sgn(�so)

mH = +�so

mH = ��so

�xy =

e2

h sgn(mH)

mK = mK0

nc = nK + nK0= 0

mK = �mK0


|q�i|q+i

⇧(q,!) =

1

V

Xk

Xnn0


Ekn � Ek�qn0 � !(1)

�Ekn 'Xn0k0

g2q


(2)

⇢q =

Xnn0

Xk,k0


�h = mS�z

�h = mH�z⌧z


n� =

12⇡

R1 dS · Fq� =

12 sgn(mvxvy)


�xx = �yy = 0

�xy =

e2

h

Xn2occ

1

2⇡

ZBZ

dS · Fkn (4)

�fkn

�| kni

1


Ion Garate

May 2, 2014

�xy =

e2

h sgn(mH)

mK = mK0

nc = nK + nK0= 0

mK = �mK0


|q�i|q+i

⇧(q,!) =

1

V

Xk

Xnn0


Ekn � Ek�qn0 � !(1)

�Ekn 'Xn0k0

g2q


(2)

⇢q =

Xnn0

Xk,k0


�h = mS�z

�h = mH�z⌧z


n� =

12⇡

R1 dS · Fq� =

12 sgn(mvxvy)


�xx = �yy = 0

�xy =

e2

h

Xn2occ

1

2⇡

ZBZ

dS · Fkn (4)

�fkn

�| kni

j = eXkn

h kn|v| knifkn (5)

En = !c(n + 1/2)

H =

12m

hp2

x +

�py � eB

c x�2

i

1

Spin-up electrons:


Ion Garate

May 2, 2014

n" = sgn(�so)

mH = +�so

mH = ��so

�xy =

e2

h sgn(mH)

mK = mK0

nc = nK + nK0= 0

mK = �mK0


|q�i|q+i

⇧(q,!) =

1

V

Xk

Xnn0


Ekn � Ek�qn0 � !(1)

�Ekn 'Xn0k0

g2q


(2)

⇢q =

Xnn0

Xk,k0


�h = mS�z

�h = mH�z⌧z


n� =

12⇡

R1 dS · Fq� =

12 sgn(mvxvy)


�xx = �yy = 0

�xy =

e2

h

Xn2occ

1

2⇡

ZBZ

dS · Fkn (4)

�fkn

�| kni

j = eXkn

h kn|v| knifkn (5)

1


Ion Garate

May 2, 2014

n" = sgn(�so)

mH = +�so

mH = ��so

�xy =

e2

h sgn(mH)

mK = mK0

nc = nK + nK0= 0

mK = �mK0


|q�i|q+i

⇧(q,!) =

1

V

Xk

Xnn0


Ekn � Ek�qn0 � !(1)

�Ekn 'Xn0k0

g2q


(2)

⇢q =

Xnn0

Xk,k0


�h = mS�z

�h = mH�z⌧z


n� =

12⇡

R1 dS · Fq� =

12 sgn(mvxvy)


�xx = �yy = 0

�xy =

e2

h

Xn2occ

1

2⇡

ZBZ

dS · Fkn (4)

�fkn

�| kni

j = eXkn

h kn|v| knifkn (5)

1

Total Chern number:

Two copies of a Haldane/Chern insulator

Haldane/Chern insulator with mass

Spin-down electrons:

Haldane/Chern insulator with mass

“Spin Chern number”:

Spin-Hall conductivity:

Quantum spin-Hall insulator

k


Ion Garate

May 2, 2014

�xy =

e2

h sgn(mH)

mK = mK0

nc = nK + nK0= 0

mK = �mK0


|q�i|q+i

⇧(q,!) =

1

V

Xk

Xnn0


Ekn � Ek�qn0 � !(1)

�Ekn 'Xn0k0

g2q


(2)

⇢q =

Xnn0

Xk,k0


�h = mS�z

�h = mH�z⌧z


n� =

12⇡

R1 dS · Fq� =

12 sgn(mvxvy)


�xx = �yy = 0

�xy =

e2

h

Xn2occ

1

2⇡

ZBZ

dS · Fkn (4)

�fkn

�| kni

j = eXkn

h kn|v| knifkn (5)

En = !c(n + 1/2)

H =

12m

hp2

x +

�py � eB

c x�2

i

1


Ion Garate

May 2, 2014

�xy =

e2

h sgn(mH)

mK = mK0

nc = nK + nK0= 0

mK = �mK0


|q�i|q+i

⇧(q,!) =

1

V

Xk

Xnn0


Ekn � Ek�qn0 � !(1)

�Ekn 'Xn0k0

g2q


(2)

⇢q =

Xnn0

Xk,k0


�h = mS�z

�h = mH�z⌧z


n� =

12⇡

R1 dS · Fq� =

12 sgn(mvxvy)


�xx = �yy = 0

�xy =

e2

h

Xn2occ

1

2⇡

ZBZ

dS · Fkn (4)

�fkn

�| kni

j = eXkn

h kn|v| knifkn (5)

En = !c(n + 1/2)

H =

12m

hp2

x +

�py � eB

c x�2

i

1


Ion Garate

May 4, 2014




�⇡/a ⇡/a h(k) = dx(k)�x+ dy(k)�y

Fn(R) = rR ⇥An(R)

�P =

e

2⇡

Xn2occ

IC

An(R) · dR =

e

2⇡

Xn2occ

ZS



Q =

Z ⌧

0

dt@P

@t= P (t = ⌧)� P (t = 0) ⌘ �P (2)

j = @P/@t⇢ = �@xP� = P · n14⇡

HS dS = 1� gH


nChern =

12⇡

RSFn(R) · dS


An(R) · dR =

RS

dS · [rR ⇥An(R)]


RC


1


Ion Garate

May 4, 2014




�⇡/a ⇡/a h(k) = dx(k)�x+ dy(k)�y

Fn(R) = rR ⇥An(R)

�P =

e

2⇡

Xn2occ

IC

An(R) · dR =

e

2⇡

Xn2occ

ZS



Q =

Z ⌧

0

dt@P

@t= P (t = ⌧)� P (t = 0) ⌘ �P (2)

j = @P/@t⇢ = �@xP� = P · n14⇡

HS dS = 1� gH


nChern =

12⇡

RSFn(R) · dS


An(R) · dR =

RS

dS · [rR ⇥An(R)]


RC


1

Spin-up edge state

Spin-down edge state

Time-reversal symmetry à (i) spin-up and spin-down edge states counter-propagate. (ii) edge states are degenerate at k=0 (Kramer’s degeneracy).

Edge states


Ion Garate

May 27, 2014

�sxy = �"

xy � �#xy = (n" � n#) e2

hnChern = nK + nK0

nChern = 0

nChern = sgn(mH)

he↵(k) ⌘ �G�1(0,k) = h(k) + ⌃(0,k)

@|E⇤g |/@T

@!⇤q/@|Eg| > 0

@!⇤q/@|Eg| < 0

g2q |h0n|qn0i|2

�E0n =

Xn0q


(1)

E⇤g 6= 2m⇤


� = 1

d|Eg|/dT < 0

d|Eg|/dT > 0

0nkn0

�so < 1mK

nK = nK0

nK = �nK0


|+i|�inK =

12 sgn(vx

KvyKmK)



h(d) = d · �|±iE± = ±|d|

1

One can no longer think of two decoupled copies of a Chern insulator. Likewise, the spin-Chern number is ill-defined.

However, the Kane-Mele edges states remain robust so long as the bulk gap does not close. Symmetry-protected topological phase.

Fundamental issue:

Practical issue:

In general spin is not conserved along any direction


Ion Garate

May 5, 2014

�so < 1mK

nK = nK0

nK = �nK0


|+i|�inK =

12 sgn(vx

KvyKmK)




�⇡/a ⇡/a h(k) = dx(k)�x+ dy(k)�y

Fn(R) = rR ⇥An(R)

�P =

e

2⇡

Xn2occ

IC

An(R) · dR =

e

2⇡

Xn2occ

ZS



R = (k, t)

Q =

Z ⌧

0

dt@P

@t= P (t = ⌧)� P (t = 0) ⌘ �P (2)

j = @P/@t⇢ = �@xP� = P · n14⇡

HS

dS = 1� gHS


nChern =

12⇡

RSFn(R) · dS


An(R) · dR =

RS

dS · [rR ⇥An(R)]

1

Alternative systems : HgTe/CdTe, InAs/GaSb, silicene,…

A (non-Chern) topological invariant is responsible for this robustness. This is the Z2 invariant. Its inception and development in 2D and 3D crystals (2005-2007) have led to an explosion of research in topological insulators.

Topological invariants in interacting systems So far, we have discussed the band topology in terms of the single-particle wave functions.

Topological Hamiltonian

How to capture the band topology of interacting systems beyond the mean field approximation?

Full Green’s function (zero Matsubara frequency)

Electron self-energy (zero Matsubara frequency)

Use the eigenstates of to calculate the Chern number and Z2 invariant in interacting systems.


Ion Garate

May 26, 2014

he↵(k) ⌘ �G�1(0,k) = h(k) + ⌃(0,k)

@|E⇤g |/@T

@!⇤q/@|Eg| > 0

@!⇤q/@|Eg| < 0

g2q |h0n|qn0i|2

�E0n =

Xn0q


(1)

E⇤g 6= 2m⇤


� = 1

d|Eg|/dT < 0

d|Eg|/dT > 0

0nkn0

�so < 1mK

nK = nK0

nK = �nK0


|+i|�inK =

12 sgn(vx

KvyKmK)




�⇡/a ⇡/a h(k) = dx(k)�x+ dy(k)�y

Fn(R) = rR ⇥An(R)

1

Non-interacting Hamiltonian

Documents

Topological Insulators: Some Basic Concepts